%  submitted to
%
%=======================================================================
%    Manuscript:
%    Title: Open Descendants of NAHE-based free fermionic and
%        Type I Z2^n models
%    Authors: Dave J. Clements and Alon E. Faraggi
%=======================================================================
%    -- submitted for publication in Physical Review D  --
%    --
%    -- no figures, no tables ---
%=======================================================================
%
%    Please address all correspondence to the author
%    at the address below:
%
%         Dr. Alon E. Faraggi
%         Theoretical Physics Department
%     University of Oxford
%         Oxford OX1 3NP
%     United Kingdom
%
%     Tel: 44 01865 273960
%     fax: 44 01865 273947
%     faraggi@thphys.ox.ac.uk
%
%    --->  Please acknowledge receipt of this manuscript
%          by e-mail as soon as possible.
%
%                          -- Thank you very much.
%=====================================================================
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%                                                                       %
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%                                                                       %
%                Dave J. Clements and Alon E. Faraggi                   %
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\begin{document}


\begin{titlepage}
\samepage{
\setcounter{page}{1}
\rightline{OUTP--03--04P}
\rightline{\tt hep-th/0302006}
\rightline{May 2002}
\vfill
\begin{center}
 {\Large \bf Open Descendants of \\NAHE--based free fermionic and Type I
$\bbb{Z}_2^n$ models\\} \vfill \vfill {\large
    David J. Clements\footnote{david.clements@new.ox.ac.uk}
        and
        Alon E. Faraggi\footnote{faraggi@thphys.ox.ac.uk}}\\
\vspace{.12in}
{\it Theoretical Physics Department, University of Oxford,
Oxford OX1 3NP\\}
\vspace{.025in}
\end{center}
\vfill
\begin{abstract}

The NAHE--set, that underlies the realistic free fermionic models,
corresponds to $\bb{Z}_2\times \bb{Z}_2$ orbifold at an enhanced
symmetry point, with $(h_{11},h_{21})=(27,3)$. Alternatively, a
manifold with the same data is obtained by starting with a
$\bb{Z}_2\times \bb{Z}_2$ orbifold at a generic point on the
lattice and adding a freely acting $\bb{Z}_2$ involution. In this
paper we study type I orientifolds on the manifolds that underly
the NAHE--based models by incorporating such freely acting shifts.
We present new models in the Type I vacuum which are modulated by
$\bb{Z}_2^n$ for $n=2,3$.  In the case of $n=2$, the
$\bb{Z}_2\times \bb{Z}_2$ structure is a composite orbifold Kaluza
Klein shift arrangement. The partition function provides a simpler
spectrum with chiral matter.  For $n=3$, the two cases discussed
are $\bb{Z}_2$ modulations of the $T^6/\bb{Z}_2 \times \bb{Z}_2$
spectrum.  The additional projection shows an enhanced closed and
open sector with chiral matter.  The second example involves a
modulation using a product of Kaluza Klein and winding shifts. The
winding shift provides this model with a simpler torus spectrum,
particularly with the removal of part of the twisted sector.  In
each of the cases the brane stacks are correspondingly altered
from those which are present in the $\bb{Z}_2\times \bb{Z}_2$
orbifold. The last two incorporate twisted terms that can provide
discrete torsion, we discuss the spectral content of each model
with discrete torsion with particular choice $\epsilon=(1,1,-1)$.



\end{abstract}
\smallskip}
\end{titlepage}

\setcounter{footnote}{0}

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%========================================================================
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%=======================================================================
\hyphenation{su-per-sym-met-ric non-su-per-sym-met-ric}
\hyphenation{space-time-super-sym-met-ric}
\hyphenation{mod-u-lar mod-u-lar--in-var-i-ant}
%=====================================================================

%============================== SECTION 1 ============================

\setcounter{footnote}{0}

\section{Introduction}

Important progress has been achieved in recent years in the basic
understanding of string theory. It is now believed that the
different string theories in ten dimensions, together with eleven
dimensional supergravity, are limits of a single more fundamental
theory, traditionally called M--theory. The question remains,
however, how to relate these advances to experimental data. In
this context some efforts have been directed at the construction
of phenomenologically viable type I string vacua \cite{typeiphun},
and nonperturbative M--theory vacua based on compactifications of
11 dimensional supergravity on CY$\times S_1/\bb{Z}_2$
\cite{horavawit,donagi,fgi} or on manifolds with $G_2$ holonomy
\cite{g2mani}.

These perturbative string constructions, however, do not yet
utilize the new M--theory picture of string theories.
The question remains how to employ this new
understanding for phenomenological studies.
In the context of M--theory
the true fundamental theory of nature should have some
nonperturbative realization. However, at present all we know
about this more basic theory are its perturbative string limits.
Therefore, we should regard these theories as
providing tools to probe the properties of the fundamental nonperturbative
vacuum in the different limits.
Each of the perturbative string
limits may therefore exhibit some properties of the
true vacuum, but it may well be that none
can characterize the vacuum completely.
In this view it is likely that all limits
will need to be used to isolate the true M--theory
vacuum. In this respect it may well
be that different perturbative string limits may provide more useful
means to study different properties of the true nonperturbative vacuum.
This suggests the following
approach to exploration of M--theory phenomenology. Namely, the
true M--theory vacuum has some nonperturbative realization that at
present we do not know how to formulate. This vacuum is at finite
coupling and is located somewhere in the space of M--theory
vacua. The properties of the true vacuum can
however be probed in the perturbative string limits. We may
hypothesize that in any of these limits one still needs to compactify
to four dimensions. Namely, that the true M--theory vacuum
can still be formulated with four non--compact and all the other
dimensions are compact. Suppose then that in some of the limits
we are able to identify a specific class of compactifications
that possess appealing phenomenological properties. The new
M--theory picture suggests that we can then explore the possible
properties of the M--theory vacuum by studying compactifications
of the other perturbative string limits on the same class of
compactifications.

In particular, we can probe those properties that pertain to the
observed experimental and cosmological data, and by using the
low energy effective field theory parameterization.
One of these properties, indicated
by the observed data, is the embedding
of the Standard Model matter states in the chiral ${\bf 16}$
representation
of $SO(10)$.
Thus, we may demand the existence of
a viable perturbative string limit which preserve this embedding.
The only perturbative string limit which enables the $SO(10)$
embedding of the Standard Model spectrum is the heterotic $E_8\times E_8$
string. The reason being that only this limit produces the
spinorial 16 representation in the perturbative massless spectrum.
Therefore, if we would like to preserve the $SO(10)$ embedding of the
Standard Model spectrum, the M--theory limit which we should use is
the perturbative heterotic string \cite{heterotic}.
In this respect it may well
be that other perturbative string limits may provide more useful
means to study different properties of the true nonperturbative vacuum,
such as dilaton and moduli stabilization \cite{casasabel}.

Pursuing this point of view, a class of realistic string models
that preserve the $SO(10)$ embedding of the Standard Model
spectrum are the NAHE--based free fermionic models. This
formulation enables detailed studies at fixed points in the moduli
space, and the models under consideration correspond to
$\bb{Z}_2\times \bb{Z}_2$ orbifold compactifications with
additional Wilson lines\footnote{It is in general anticipated that
the different formulations of string compactifications to four
dimensions do not represent different physics and are related,
even if the dictionary is not always known.}. Many of the
encouraging phenomenological characteristics of the realistic free
fermionic models are rooted in the underlying $\bb{Z}_2\times
\bb{Z}_2$ orbifold structure, including the three generations
arising from the three twisted sectors, and the canonical SO(10)
embedding for the weak hyper-charge. We may therefore regard the
phenomenological success of the free fermionic models as
highlighting a specific class of compactified manifolds.

Given the specific class of compactified manifolds highlighted by
NAHE--based free fermionic models, the line of approach
to phenomenological studies of M--theory
that we pursue here is to compactify
other perturbative string limits on the same manifolds. It is then
hoped that these studies will elucidate other properties of these
realistic models. This is the line of thought that was pursued in
ref. \cite{fgi} where compactification of Horava--Witten theory to
four dimensions on manifolds that are related to the free
fermionic models were studied.

Pursuing this approach we study in this paper orientifolds of type
IIB string theory on the manifolds that are related to the free
fermionic models. The geometric manifold that underlies the free
fermionic models is a $\bb{Z}_2\times \bb{Z}_2$ orbifold at an
enhanced symmetry point in the Narain moduli space. At the free
fermionic point the Narain lattice arising from the six
compactified dimensions is enhanced from $U(1)^6$ to $SO(12)$. The
$\bb{Z}_2\times \bb{Z}_2$ orbifold projection of this lattice then
yields a manifold with $(h_{11},h_{21})=(27,3)$. On the other hand
a $\bb{Z}_2\times \bb{Z}_2$ orbifold projection at a generic point
in the moduli space yields a manifold with
$(h_{21},h_{11})=(51,3)$. We refer to the later as $X_1$ and to
the former as $X_2$. These two manifolds can alternatively be
connected by adding a freely acting shift to $X_1$, which reduces
the number of twisted fixed points by 1/2. Orientifolds of
$\bb{Z}_2\times \bb{Z}_2$ orbifolds were studied in ref.
\cite{z2z2orient}. To advance these studies toward nonperturbative
studies of the free fermionic models we therefore extend this
analysis by including the freely acting shift that connects the
$X_1$ and $X_2$ manifolds.

\setcounter{footnote}{0}
\section{Realistic free fermionic models - general structure}\label{gs}
In this section we recapitulate the main structure of
the realistic free fermionic models.
The notation
and details of the construction of these
models are given elsewhere \cite{rffm}.
In the free fermionic formulation \cite{fff} of the heterotic string
in four dimensions a model is specified in terms of boundary
condition basis vectors and one--loop GSO phases.
The physical spectrum is obtained by applying the generalized GSO projections.
The boundary condition basis defining a typical
realistic free fermionic heterotic string models is
constructed in two stages.
The first stage consists of the NAHE set,
which is a set of five boundary condition basis vectors,
$\{{\bf1},S,b_1,b_2,b_3\}$ \cite{nahe}.
The gauge group after imposing the GSO projections induced
by the NAHE set is $SO(10)\times SO(6)^3\times E_8$
with $N=1$ supersymmetry.
At the level of the NAHE set the sectors $b_1$, $b_2$ and $b_3$
produce 48 multiplets, 16 from each, in the $16$
representation of $SO(10)$. The states from the sectors $b_j$
are singlets of the hidden $E_8$ gauge group and transform
under the horizontal $SO(6)_j$ $(j=1,2,3)$ symmetries.
This structure is common to all the realistic free fermionic models.

The second stage of the
basis construction consists of adding to the
NAHE set three (or four) additional boundary condition basis vectors,
typically denoted by $\{\alpha,\beta,\gamma\}$.
These additional basis vectors reduce the number of generations
to three chiral generations, one from each of the sectors $b_1$,
$b_2$ and $b_3$, and simultaneously break the four dimensional
gauge group. The assignment of boundary conditions
breaks $SO(10)$ to one of its subgroups \cite{rffm}.
Similarly, the hidden $E_8$ symmetry is broken to one of its
subgroups by the basis vectors which extend the NAHE set.
The flavor $SO(6)^3$ symmetries in the NAHE--based models
are broken to flavor $U(1)$ symmetries.
The three additional basis vectors $\{\alpha, \beta, \gamma\}$ differ
between the models and there exists a large number of viable three
generation models in this class.

 From the preceding discussion it follows that the underlying
$\bb{Z}_2\times \bb{Z}_2$ orbifold structure is common to all the
three generation free fermionic models. This is the structure that
we will exploit in trying to elevate the study of these models
across the strong--weak duality barrier. In this respect our aim
is to explore which of the structures of these models is preserved
in the nonperturbative domain. We should note that at this stage
the analysis in this respect is purely exploratory.

The correspondence of the NAHE--based free fermionic models
with the orbifold construction is illustrated
by extending the NAHE set, $\{{\bf1},S,b_1,b_2,b_3\}$, by one additional
boundary condition basis vector \cite{foc}, $\xi_1$.
With a suitable choice of the GSO projection coefficients the
model possess an $SO(4)^3\times E_6\times U(1)^2\times E_8$ gauge group
and $N=1$ space--time supersymmetry. The matter fields
include 24 generations in the 27 representations of
$E_6$, eight from each of the sectors $b_1\oplus b_1+\xi_1$,
$b_2\oplus b_2+\xi_1$ and $b_3\oplus b_3+\xi_1$.
Three additional 27 and $\overline{27}$ pairs are obtained
from the Neveu--Schwarz $\oplus~\xi_1$ sector.

To construct the model in the orbifold formulation one starts
with a model compactified on a flat torus with nontrivial background
fields \cite{Narain}.
The subset of basis vectors
\beq
\{{\bf1},S,\xi_1,\xi_2\},
\label{neq4set}
\eeq
with $\xi_2={\bf1}+b_1+b_2+b_3$,
generates a toroidally-compactified model with $N=4$ space--time
supersymmetry and $SO(12)\times E_8\times E_8$ gauge group.
The same model is obtained in the geometric (bosonic) language
by constructing the background fields which produce
the $SO(12)$ lattice. The metric of the six-dimensional compactified
manifold is taken as the Cartan matrix of $SO(12)$,
and the antisymmetric tensor is given by
\beq
B_{ij}=\cases{
G_{ij}&;\ $i>j$,\cr
0&;\ $i=j$,\cr
-G_{ij}&;\ $i<j$.\cr}
\label{bso12}
\eeq
When all the radii of the six-dimensional compactified
manifold are fixed at $R_I=\sqrt2$, it is seen that the
left-- and right--moving momenta
$%\beq
P^I_{R,L}=[m_i-{1\over2}(B_{ij}{\pm}G_{ij})n_j]{e_i^I}^*
%\label{lrmomenta}
$%\eeq
reproduce all the massless root vectors in the lattice of
$SO(12)$. Here $e^i=\{e_i^I\}$ are six linearly-independent
vectors normalized: $(e_i)^2=2$.
The ${e_i^I}^*$ are dual to the $e_i$, with
$e_i^*\cdot e_j=\delta_{ij}$.

Adding the two basis vectors $b_1$ and $b_2$ to the set
(\ref{neq4set}) corresponds to the $\bb{Z}_2\times \bb{Z}_2$
orbifold model with standard embedding. Starting from the Narain
model with $SO(12)\times E_8\times E_8$ symmetry~\cite{Narain},
and applying the $\bb{Z}_2\times \bb{Z}_2$ twisting on the
internal coordinates, reproduces the spectrum of the free-fermion
model with the six-dimensional basis set
$\{{\bf1},S,\xi_1,\xi_2,b_1,b_2\}$. The Euler characteristic of
this model is 48 with $h_{11}=27$ and $h_{21}=3$.

It is noted that the effect of the additional basis vector $\xi_1$
is to separate the gauge degrees of freedom {}from the internal
compactified degrees of freedom. In the realistic free fermionic
models this is achieved by the vector $2\gamma$ \cite{foc}, which
breaks the $E_8\times E_8$ symmetry to $SO(16)\times SO(16)$. The
$\bb{Z}_2\times \bb{Z}_2$ twisting breaks the gauge symmetry to
$SO(4)^3\times SO(10)\times U(1)^3\times SO(16)$. The orbifold
twisting still yields a model with 24 generations, eight from each
twisted sector, but now the generations are in the chiral 16
representation of $SO(10)$, rather than in the 27 of $E_6$. The
same model can be realized with the set
$\{{\bf1},S,\xi_1,\xi_2,b_1,b_2\}$, by projecting out the
$16\oplus{\overline{16}}$ from the sector $\xi_1$ by taking \beq
c{\xi_1\choose \xi_2}\rightarrow -c{\xi_1\choose \xi_2}.
\label{changec} \eeq This choice also projects out the massless
vector bosons in the 128 of $SO(16)$ in the hidden-sector $E_8$
gauge group, thereby breaking the $E_6\times E_8$ symmetry to
$SO(10)\times U(1)\times SO(16)$. The freedom in eq.
({\ref{changec}) correspond to a discrete torsion in the toroidal
orbifold model. At the level of the $N=4$ Narain model generated
by the set (\ref{neq4set}), we can define two models, $Z_+$ and
$Z_-$, depending on the sign of the discrete torsion in eq.
(\ref{changec}). One model, say $Z_+$, produces the $E_8\times
E_8$ model, whereas the second, say $Z_-$, produces the
$SO(16)\times SO(16)$ model. However, the $\bb{Z}_2\times
\bb{Z}_2$ twists act identically in the two models, and their
physical characteristics differ only due to the discrete torsion
eq. (\ref{changec}).

This analysis confirms that the $\bb{Z}_2\times \bb{Z}_2$ orbifold
on the $SO(12)$ Narain lattice is indeed at the core of the
realistic free fermionic models. However, this orbifold model
differs from the $\bb{Z}_2\times \bb{Z}_2$ orbifold on
$T_2^1\times T_2^2\times T_2^3$ with $(h_{11},h_{21})=(51,3)$. In
ref. \cite{befnq} it was shown that the two models are connected
by adding a freely acting twist or shift to the $X_1$ model. Let
us first start with the compactified $T^1_2\times T^2_2\times
T^3_2$ torus parameterized by three complex coordinates $z_1$,
$z_2$ and $z_3$, with the identification \beq z_i=z_i +
1~~~~~~~~~~;~~~~~~~~~~z_i=z_i+\tau_i \label{t2cube} \eeq where
$\tau$ is the complex parameter of each $T_2$ torus. With the
identification $z_i\rightarrow-z_i$, a single torus has four fixed
points at \beq z_i=\{0,1/2,\tau/2,(1+\tau)/2\}. \label{fixedtau}
\eeq With the two $\bb{Z}_2$ twists \beqn &&
\alpha:(z_1,z_2,z_3)\rightarrow(-z_1,-z_2,~~z_3)\cr &&
\beta:(z_1,z_2,z_3)\rightarrow(~~z_1,-z_2,-z_3), \label{alphabeta}
\eeqn there are three twisted sectors in this model, $\alpha$,
$\beta$ and $\alpha\beta=\alpha\cdot\beta$, each producing 16
fixed tori, for a total of 48. Adding to the model generated by
the $\bb{Z}_2\times \bb{Z}_2$ twists in (\ref{alphabeta}), the
additional shift \beq
\gamma:(z_1,z_2,z_3)\rightarrow(z_1+{1\over2},z_2+{1\over2},z_3+{1\over2})
\label{gammashift} \eeq produces again a fixed tori from the three
twisted sectors $\alpha$, $\beta$ and $\alpha\beta$. The product
of the $\gamma$ shift in (\ref{gammashift}) with any of the
twisted sectors does not produce any additional fixed tori.
Therefore, this shift acts freely. Under the action of the
$\gamma$ shift, half the fixed tori from each twisted sector are
paired. Therefore, the action of this shift is to reduce the total
number of fixed tori from the twisted sectors by a factor of
$1/2$, with $(h_{11},h_{21})=(27,3)$. This model therefore
reproduces the data of the $\bb{Z}_2\times \bb{Z}_2$ orbifold at
the free-fermion point in the Narain moduli space.

We noted above that the freely acting shift (\ref{gammashift}),
added to the ${\bb{Z}}_2\times {\bb{Z}}_2$ orbifold at a generic
point of $T_2^1\times T_2^2\times T_2^3$, reproduces the data of
the ${\bb{Z}}_2\times {\bb{Z}}_2$ orbifold acting on the SO(12)
lattice. This observation does not prove, however, that the vacuum
which includes the shift is identical to the free fermionic model.
While the massless spectrum of the two models may coincide their
massive excitations, in general, may differ. The matching of the
massive spectra is examined by constructing the partition function
of the ${\bb{Z}}_2\times {\bb{Z}}_2$ orbifold of an SO(12)
lattice, and subsequently that of the model at a generic point
including the shift. In effect since the action of the
${\bb{Z}}_2\times {\bb{Z}}_2$ orbifold in the two cases is
identical the problem reduces to proving the existence of a freely
acting shift that reproduces the partition function of the SO(12)
lattice at the free fermionic point. Then since the action of the
shift and the orbifold projections are commuting it follows that
the two ${\bb{Z}}_2\times {\bb{Z}}_2$ orbifolds are identical.

On the compact coordinates there are actually three inequivalent ways
in which the shifts
can act. In the more familiar case, they simply translate a generic point
by half the
length of the circle. As usual, the presence of windings in string
theory allows shifts on the T-dual circle, or even asymmetric ones, that
act both on the circle and on its dual. More concretely, for a circle of
length $2 \pi R$, one can have the following possibilities \cite{vwaaf}:
\beqn
A_1\;:&& X_{\rm L,R} \to X_{\rm L,R} + {\textstyle{1\over 2}} \pi R \,,
\nonumber \\
A_2\;:&& X_{\rm L,R} \to X_{\rm L,R} + {\textstyle{1\over 2}} \left(
\pi R \pm {\pi \alpha ' \over R} \right) \,,
\nonumber \\
A_3\;:&& X_{\rm L,R} \to X_{\rm L,R} \pm {\textstyle{1\over 2}}
{\pi \alpha' \over R} \,. \label{a1a2a3} \eeqn There is an
important difference between these choices: while $A_1$ and $A_3$
can act consistently on any number of coordinates, level-matching
requires instead that $A_2$ acts on (mod) four real coordinates.
By studying the respective partition function one finds
\cite{partitions} that the shift that reproduces the $SO(12)$
lattice at the free fermionic point in the moduli space is
generated by the ${\bb{Z}}_2\times {\bb{Z}}_2$ shifts \beqn g\;: &
& (A_2 , A_2 ,0 ) \,,
\nonumber \\
h\;: & & (0, A_2 , A_2 ) \,, \label{gfh} \eeqn where each $A_2$
acts on a complex coordinate. It is then shown that the partition
function of the SO(12) lattice is reproduced. at the self-dual
radius, $R_i = \sqrt{\alpha '}$. On the other hand, the shifts
given in Eq. (\ref{gammashift}), and similarly the analogous
freely acting shift given by $(A_3,A_3,A_3)$, do not reproduce the
partition function of the $SO(12)$ lattice. Therefore, the shift
in eq. (\ref{gammashift}) does reproduce the same massless
spectrum and symmetries of the ${\bb{Z}}_2\times {\bb{Z}}_2$ at
the free fermionic point, but the partition functions of the two
models differ! Thus, the free fermionic ${\bb{Z}}_2\times
{\bb{Z}}_2$ is realized for a specific form of the freely acting
shift given in eq. (\ref{gfh}). However, all the models that are
obtained from $X_1$ by a freely acting ${\bb{Z}}_2$-shift have
$(h_{11},h_{21})=(27,3)$ and hence are connected by continuous
extrapolations. The study of these shifts in themselves may
therefore also yield additional information on the vacuum
structure of these models and is worthy of exploration.

Despite its innocuous appearance the connection between $X_1$ and
$X_2$ by a freely acting shift has the profound consequence of
making the $X_2$ manifold non--simply connected, which allows the
breaking of the SO(10) symmetry to one of its subgroups. Thus, we
can regard the utility of the free fermionic machinery as singling
out a specific class of $\bb{Z}_2\times \bb{Z}_2$ compactified
manifolds. In this context the freely acting shift has the crucial
function of connecting between the simply connected covering
manifold to the non-simply connected manifold. Precisely such a
construction has been utilized in \cite{donagi,fgi} to construct
non-perturbative vacua of heterotic M-theory. In the next section
we turn to study open descendants of $\bb{Z}_2\times \bb{Z}_2$
orbifolds that incorporate such freely acting shifts.

\setcounter{footnote}{0}

%%%%%%%%%%%% New Section %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{$\bbbb{Z}_2\times \bbbb{Z}_2$ Model With Composite Shift Orbifold
Generators}

%%%%%%%%%%%% New Section %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

To illustrate the effects of the freely acting shifts of the type
in eq. (\ref{gammashift}) on the open descendants, we start with a
simpler example of a $\bb{Z}_2$ orbifold, $g$ and an additional
freely acting shift $h$. The action of $g$ and $h$ and their
products is given in eq.
(\ref{eqn:CompositeGenerators})\footnote{This model was analyzed
in collaboration with Carlo Angelantonj and Emilian Dudas.}.

The $\bb{Z}_2\times \bb{Z}_2$ generators have both an action on
the string coordinates (as a parity projection), and the topology
of the internal directions, in that they break $T^6$ to
$T^2_{45}\times T^2_{67}\times T^2_{89}$, with subscripts
referring to the 2-tori directions.  As such, the original type
IIB theory is projected using
%
%
\begin{eqnarray}\label{eqn:CompositeGenerators}
g &=& (1~,~1;-1~,-1;-1,-1~),\nn\\
h &=& (A_1,1;~A_1~,~1~;~~1~,~1~), \nn\\
f &=& (A_1,1;-A_1,-1;-1,-1),
\end{eqnarray}
%
%
for $A_1$ defined in (\ref{a1a2a3}).  The generators,
(\ref{eqn:CompositeGenerators}) illustrates the shift action on
only one of the coordinates of the relevant torus. The orbifolds
act on all coordinates within a given torus to provide four fixed
points.

This is an interesting model that has a $\bb{Z}_2\times \bb{Z}_2$
structure while preserving ${\cal N}= 2$ supersymmetry. The choice
of generators that has at least two with shift operators, has the
effect of shifting elements of a matrix $M$ (which encodes the
positions of the fixed points)
\begin{eqnarray}\label{eqn:MatrixM}
Tr h Mq^{L_0}q^{\bar{L}_0}
\end{eqnarray}
to the off diagonal positions in the torus amplitude. This implies
that the independent orbit diagrams (those not related by modular
transformation) no longer contribute to the torus
amplitude. This takes away the consideration of a sub class of
models associated with discrete torsion, which corresponds to such
independent orbits, and thus neglects supersymmetry breaking from
the inclusion of antibranes.  As will be shown, sign changes
arising from the discrete torsion terms will necessarily change
the charge of the brane that they couple to.

The way the modulating group generators are written with composite
shift operators, has a twofold effect, firstly it will necessarily
force the number of distinct $D5$ embedding types to become only
one (in this case, the first torus will provide the $D5$ physics).
This happens because the lifting of lattice states forces the
tadpole condition to eliminate the corresponding brane.  Secondly,
the particular arrangement of the shifted directions will allow
for an interesting geometrical configuration of the $O$-planes in
the Klein amplitude.

{}From (\ref{eqn:CompositeGenerators}), it is appreciated that the
vacuum is left with 2 degrees of freedom in the $R$-$R$ sector by
virtue of the $h$ generator, since it does not make a further
connection (outside that which is provided by the orbifold on the
$45$ and $67$ directions) between the $SO(2)^4$ vacua. The model
thus has an ${\cal N}= 2$ supersymmetry in the bulk.

The ${\cal N}=2$ character set is derived from the breaking of the
original SO(8) lightcone characters $O_8$, $V_8$, $C_8$ and $S_8$
to supersymmetric representations involving $O_4$, $V_4$, $C_4$
and $S_4$.  The type I constructions are discussed in detail in \cite{CAAS}.
In this case, the supersymmetric fermion contributions are written
as
\begin{table}[!h]
\begin{center}
\begin{tabular}{ll}
$Q_o=V_4O_4-C_4C_4$, & $Q_v=O_4V_4-S_4S_4$ \\
$Q_s=O_4C_4-S_4O_4$, & $Q_c=V_4S_4-C_4V_4$.
\end{tabular}
\end{center}
\end{table}

The $SO(2n)$ characters are,
\begin{table}[!h]
\begin{center}
\begin{tabular}{ll}
$O_{2n}=\frac{1}{2\eta^n}\big( \theta^n_3+\theta^n_4
\big)$, &
$V_{2n}=\frac{1}{2\eta^n}\big( \theta^n_3-\theta^n_4 \big)$, \\
$S_{2n}=\frac{1}{2\eta^n}\big( \theta^n_2+i^{-n}\theta^n_1 \big)$
& $C_{2n}=\frac{1}{2\eta^n}\big( \theta^n_2-i^{-n}\theta^n_1
\big)$.
\end{tabular}
\end{center}
\end{table}\label{eqn:exp}\\
with the Dedekind eta function
\begin{eqnarray}
\eta=q^{\frac{1}{24}}\prod_{n=1}^{\infty}\big(1-q^n\big),\nn
\end{eqnarray}
their respective conformal weights are $0$, $\frac{1}{2}$,
$\frac{n}{8}$ and $\frac{n}{8}$. Here, these are representations
of a scalar, a vector, chiral and anti-chiral spinors. The theta
functions originate from the $NS$ and $R$ sectors and are defined
by
\begin{eqnarray}
\theta \left[ \matrix{\chi \cr \phi \cr } \right] =\sum_n
q^{\frac{1}{2}(n+\chi)^2}e^{2\pi i(n+\chi)\phi}
\end{eqnarray}
where $\chi$ and $\phi$ take the values $\frac{1}{2}$ $(NS)$ and
$0$ $(R)$, the labelled theta functions are then defined as,
\begin{eqnarray}
\theta_1 = \left[ \matrix{ \frac{1}{2} \cr \frac{1}{2} \cr }
\right], \quad \theta_2 = \left[ \matrix{ \frac{1}{2} \cr 0 \cr }
\right], \quad \theta_3 = \left[ \matrix{ 0 \cr 0 \cr }
\right]\quad{\rm and}\quad \theta_4 = \left[ \matrix{ 0 \cr
\frac{1}{2} \cr } \right].
\end{eqnarray}

We will use a compact notation for the lattice modes arising from
the compact momenta as
\begin{eqnarray}\label{eqn:compact}
\Lambda_{m+a,n+b}=
\frac{q^{\frac{\alpha\prime}{4}{\big{(}\frac{(m+a)}{R}+\frac{(n+b)R}
{\alpha\prime}\big{)}}^2}\bar{q}^{\frac{\alpha\prime}{4}
{\big{(}\frac{(m+a)}{R}-\frac{(n+b)R}{\alpha\prime}\big{)}}^2}}
{\eta(q)\eta(\bar{q})}.\nn
\end{eqnarray}

To obtain modular invariance under $SL(2,\bb{Z})$, as required by
the topology of the one loop string amplitude, one must perform
$S$ and $T$ transforms, the generators of this group, which act on
the complex torus covering parameter $\tau$ as
\begin{eqnarray}\label{eqn:tran}
S:\tau\rightarrow-\frac{1}{\tau} && \Rightarrow(a,b)\rightarrow(b,a^{-1})\nn\\
T:\tau\rightarrow\tau+1 && \Rightarrow(a,b)\rightarrow(a,a b)\nn\\
\end{eqnarray}
%
%
Here, $a$ and $b$ refer to the orbit operations that are placed on
two sides of the torus world--sheet.

\begin{table}[!ht]
\begin{center}
\begin{tabular}{|c|c|c|}\hline
Lattice & ${\cal A}$ & ${\cal K}~$and$~{\cal M}$ \\\hline $P_m$ &
$W_n$ &
$W_{2n}$\\
$(-1)^mP_{m+\frac{1}{2}}$ & $(-1)^nW_{n+\frac{1}{2}}$ &
$(-1)^nW_{2n+1}$\\
$P_{m+\frac{1}{2}}$ & $(-1)^nW_n$ & $(-1)^nW_{2n}$\\
$(-1)^mP_m$ & $W_{n+\frac{1}{2}}$ & $W_{2n+1}$ \\\hline
\end{tabular}
\caption{Lattice $S$ transforms}\label{tab:Stransforms}
\end{center}
\end{table}

Here, ${\cal A}$, ${\cal K}$ and ${\cal M}$ are the annulus Klein
and Mobius contributions, and the relevant terms can be seen by
using the appropriate form for the measure parameter in each
case\footnote{The shift in mass by applying an $S$ transformation
on terms involving a phase is illustrated in appendix
\ref{app:massshift}}.  $P$ and $W$ are the restriction of
$\Lambda_{m,n}$ to pure Kaluza-Klein ($P$) or winding ($W$) modes.
The further notation of $P_o$ and $P_e$ (and similarly for the
winding sums) are the restriction of the counting to even or odd
modes only.  In the case of odd lattices, the convention should
not be taken to be correlated with labels on the fermionic
contributions. The action on these lattice modes for $S$
transformations are as in table \ref{tab:Stransforms}.  The action
of $T$ on the lattices are
\begin{eqnarray}\label{eqn:Ttransforms}
\begin{array}{lcl}
\Lambda_{m,n}\rightarrow\Lambda_{m,n}, & &
\Lambda_{m,n+\frac{1}{2}}\rightarrow(-1)^m\Lambda_{m,n+\frac{1}{2}}, \\
\Lambda_{m+\frac{1}{2},n}\rightarrow(-1)^n\Lambda_{m+\frac{1}{2},n},
& { \rm and } & \Lambda_{m+\frac{1}{2},n+\frac{1}{2}}\rightarrow
i(-1)^{m+n}\Lambda_{m+\frac{1}{2},n+\frac{1}{2}}.\nn
\end{array}
\end{eqnarray}

Thus the modular invariant torus amplitude is
\begin{eqnarray}
{\cal
%% FOLLOWING LINE CANNOT BE BROKEN BEFORE 80 CHAR
T}=&\frac{1}{4}&\bigg\{\big[1+(-1)^{m_1+m_2}\big]
(\Lambda^1\Lambda^2+\Lambda^1_{m,n+\frac{1}{2}}
\Lambda^2_{m,n+\frac{1}{2}})|Q_o+Q_v|^2\nn\\
&&+\big[1+(-1)^{m_1}\big]\Lambda^1
+|Q_o-Q_v|^2{\vline\frac{2\eta}{\theta_2}\vline}^4\nn\\
&&+16(\Lambda^1+\Lambda^1_{m,n+\frac{1}{2}})
{\vline\frac{\eta}{\theta_4}\vline}^4|Q_s+Q_c|^2\nn\\
&&+16(\Lambda^1+(-1)^{m_1}\Lambda^1_{m,n+\frac{1}{2}})
{\vline\frac{\eta}{\theta_3}\vline}^4|Q_s-Q_c|^2\bigg\}.
\end{eqnarray}
With the $A_1$ shift operator, the number of fixed points can be
seen to be halved, it acts on the fixed point coordinates as
\begin{eqnarray}
(0,0;0,0)\rightarrow(0+\frac{1}{2},0;0+\frac{1}{2},0).
\end{eqnarray}
The total number of fixed points without the shift operation is
16, which are detailed in table (\ref{tab:fixedpoints}).
\begin{table}[!ht]
\begin{center}
\begin{tabular}{|llll|}\hline
${(0,0;0,0)}_1$ & ${(0,\frac{1}{2};0,0)}_2$ &
${(\frac{1}{2},0;0,0)}_3$ & ${(\frac{1}{2},\frac{1}{2};0,0)}_4$
\\ ${(0,0;0,\frac{1}{2})}_5$ & ${(0,0;\frac{1}{2},0)}_6$ &
${(0,0;\frac{1}{2},\frac{1}{2})}_7$ & ${(0,\frac{1}{2};0,\frac{1}{2})}_8$\\
${(0,\frac{1}{2};\frac{1}{2},0)}_9$ &
${(0,\frac{1}{2};\frac{1}{2},\frac{1}{2})}_{10}$ &
${(\frac{1}{2},0;0,\frac{1}{2})}_{11}$ &
${(\frac{1}{2},0;\frac{1}{2},0)}_{12}$\\
${(\frac{1}{2},0;\frac{1}{2},\frac{1}{2})}_{13}$ &
${(\frac{1}{2},\frac{1}{2};0,\frac{1}{2})}_{14}$ &
${(\frac{1}{2},\frac{1}{2};\frac{1}{2},0)}_{15}$ &
${(\frac{1}{2},\frac{1}{2};\frac{1}{2},\frac{1}{2})}_{16}$ \\ \hline
\end{tabular}
\caption{Unshifted fixed points}\label{tab:fixedpoints}
\end{center}
\end{table}
The origin of the lattice contributions of the torus amplitude
\begin{eqnarray}
{\cal T}_0=|Q_o|^{2}+|Q_v|^2+8|Q_s|^2+\ldots.
\end{eqnarray}
shows, as expected, 8 fixed points, reduced from 16 within a given
orbifold projection, the independent coordinates of which are as
in table \ref{tab:shiftedFixedPoints}.
\begin{table}[!ht]
\begin{center}
\begin{tabular}{|llll|}\hline
${(0,0;0,0)}_1$ & ${(0,\frac{1}{2};0,0)}_2$ &
${(\frac{1}{2},0;0,0)}_3$ &
${(\frac{1}{2},\frac{1}{2};0,0)}_4$\\
${(0,\frac{1}{2};\frac{1}{2},0)}_9$ &
${(0,\frac{1}{2};\frac{1}{2},\frac{1}{2})}_{10}$ &
${(\frac{1}{2},0;0,\frac{1}{2})}_{11}$ &
${(\frac{1}{2},0;\frac{1}{2},0)}_{12}$ \\ \hline
\end{tabular}
\caption{Identified fixed points}\label{tab:shiftedFixedPoints}
\end{center}
\end{table}

Vertex operators of states flowing in $K$ and $\tilde{{\cal A}}$
will acquire from the torus, by virtue of the action of the shift
in $T_{45}^2$ and $T_{67}^2$, a state projector
\begin{eqnarray}
V=\big[1+(-1)^{m_1+m_2}\big]V_{(T^4/\ssbb{Z}_2)\times \ssbb{Z}_2}.
\end{eqnarray}

The Klein in the direct channel must realize a proper particle
interpretation, in that the coefficients of the series expansion
of the closed string amplitudes must be integer. Such a
symmetrization becomes clear from the type IIB projection
\begin{eqnarray}
Tr_{IIB}\frac{(1+\Omega)}{2}q^{L_0}{\bar{q}}^{L_0},
\end{eqnarray}
where $\Omega$ has the usual definition of the world sheet parity
operator. The Klein corresponds to the $\Omega$ projected
contribution. Under $\Omega$, the torus states are reduced as in table
\ref{tab:OmegaReduction}.
\begin{table}[!ht]
\begin{center}
\begin{tabular}{|ll|} \hline
$|T_{oo}|^2\quad\rightarrow\quad T_{oo}$ &
$|T_{om}|^2\quad\rightarrow\quad T_{oo}$ \\
$|T_{ko}|^2\quad\rightarrow\quad T_{ko}$ &
$|T_{km}|^2\quad\rightarrow\quad T_{ko}$\\ \hline
\end{tabular}
\end{center}
\caption{$\Omega$ Reduction of Torus States}\label{tab:OmegaReduction}
\end{table}
In a similar fashion, this projection also reduces the lattice
modes to become either pure momentum or pure winding, this
situation is inverted with the assistance of an inserted orbifold
action $\alpha$:
\begin{eqnarray}
\Omega|p_L,p_R>&=&|p_R,p_L>\quad\Rightarrow\quad n=0,\nn\\
\Omega\alpha|p_L,p_R>&=&|-p_R,-p_L>\quad\Rightarrow\quad m=0.
\end{eqnarray}
If a state results from orbifold actions on the left and right
states under parity identification, the orbifold generators are
correspondingly identified and allude to a trivial action on the
Klein amplitude, which takes the form,
\begin{eqnarray}
{\cal K}=&\frac{1}{8}&\bigg\{\big[\big(1+(-1)^{m_1+m_2}\big)P_1P_2P_3\nn\\
&&+\big(1+(-1)^{m_1}\big)P_1W_2W_3\big](Q_o+Q_v)\nn\\
&&+32(Q_s+Q_c)P_1{\bigg(\frac{\eta}{\theta_4}\bigg)}^2\bigg\}
\end{eqnarray}
With corresponding transverse amplitude
\begin{eqnarray}
\tilde{{\cal K}}=&\frac{2^5}{8}&\bigg\{\big[(W_1^eW_2^e+W_1^oW_2^o)
W_3^ev_1v_2v_3+W_1P_2^eP_3^e\frac{v_1}{v_2v_3}\big](Q_o+Q_v)\nn\\
&&+2v_1W_1^e(Q_o-Q_v){\bigg(\frac{\eta}{\theta_2}\bigg)}^2)\bigg\}
\end{eqnarray}

Although the $O$--planes present are not indicated explicitly
within amplitudes, their presence and dimension are understood
from the relative charges involving the $v_i$ terms. $O9$ planes
occupy the entire compact space and so correspond to the term
$v_1v_2v_3$.  $O5$ only has a presence in the first of the three
2-tori, and so has a volume term $\frac{v_1}{v_2v_3}$.

The geometry of the O-planes here provide an interesting
realization, they arrange themselves in a diagonal manner due to
the $h$ projection, which is a pure shift. For example, by
performing 2 T-dualities along the 2 directions where the $h$
shift acts in $T_1^2$ and $T_2^2$, the dilaton wave function
\begin{eqnarray}
\phi(y_1,y_2)=\sum_{m_1,m_2}\bigg(cos\big(\frac{m_1y_1}{R_1}\big)
cos\big(\frac{m_2y_2}{R_2}\big)+sin\big(\frac{m_1y_1}
{R_1}\big)sin\big(\frac{m_2y_2}{R_2}\big)\bigg)\phi^{(m_1,m_2)}\nn
\end{eqnarray}
gives access to the positions as in (figure \ref{KFP}).

%%%%%%%%%%%%%%%%%%%%%%Klein Fixed Points%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}
\centerline{\epsfxsize 1.0 truein \epsfbox {O7.eps}}
\caption{Klein Fixed Point Orientation} \label{KFP}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%END OF FIGURE%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

By a matter of interpretation, the charges must arrange themselves
as the perfect squares.  This comes from the transverse channel that provides
a tree level coupling between two orientifolds and a closed string.  The cross
terms
then give the mixing of different orientifold types.  The same is true in the
transverse
annulus for brane couplings.

The origin of the lattices in the
transverse Klein amplitude of tori that contribute to the perfect
squares shows their form as
\begin{eqnarray}
\tilde{{\cal K}}_o&=&\frac{2^5}{8}v_1\bigg\{\big(\sqrt{v_2v_3}+
\frac{1}{\sqrt{v_2v_3}}\big)^2W_1^eQ_o+\big(\sqrt{v_2v_3}-
\frac{1}{\sqrt{v_2v_3}}\big)^2W_1^eQ_v\nn\\
&&+\bigg[v_2v_3\big(W_1^oW_2^oW_3^e+W_1^e(W_2^eW_3^e-1)\big)\nn\\
&&+\frac{1}{v_2v_3}\big(W_1^oP_2^eP_3^e+W_1^e(P_2^eP_3^e-1)\big)\bigg]
(Q_o+Q_v)\bigg\}
\end{eqnarray}
with massive states shown to illustrate proper particle
interpretation with the torus for both massive and massless closed
spectra.  The transverse annulus is derived from the states that
flow in the torus, with the restriction to winding (N) or Kaluza
Klein (D) states (with the appropriate string boundary
conditions).  This manifestly changes the counting in the form of
$D9$ or $D5$ artifacts.  The transverse amplitude is
\begin{eqnarray}\label{eqn:CompositeTransverseAnnulus}
\tilde{{\cal A}}&=&\frac{2^{-5}}{8}v_1\bigg\{\big[N^2v_2v_3(W_1W_2+
W_1^{n+\frac{1}{2}}W_2^{n+\frac{1}{2}})W_3\nn\\
&&+\frac{4D^2}{v_2v_3}W_1P^e_2P_3\big](Q_o+Q_v)+
4NDW_1(Q_o-Q_v){\bigg(\frac{2\eta}{\theta_2}\bigg)}^2\bigg\}\nn\\
&&+\frac{2^{-3}}{8}v_1\bigg\{\big[R_N^2(W_1+W_1^{n+\frac{1}{2}})+
2R_D^2W_1\big](Q_s+Q_c){\bigg(\frac{2\eta}{\theta_4}\bigg)}^2\bigg\}\nn\\
&&-4R_NR_DW_1(Q_s-Q_c){\bigg(\frac{\eta}{\theta_3}\bigg)}^2.
\end{eqnarray}
As will be shown in greater detail later with the case of
$T^6/\bb{Z}_2^3$, the various factors of the states are fixed by
the requirement of perfect square couplings between the $D5$ and
$D9$ branes and fixed point occupation numbers in the twisted
sector.

%%%%%%%%%%%%%%%%%%%%%%Transverse Annulus D5 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}
\centerline{\epsfxsize 2.0 truein \epsfbox {annulusZ2xZ2.eps}}
\caption{$D5$ and $D5^\prime$ configuration} \label{TransA}
\end{figure}
%%%%%%%%%%%%%%%%%%%%% END OF FIGURE %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The transverse states (\ref{eqn:CompositeTransverseAnnulus}) then
highlight the $D5$ orientation as in figure \ref{TransA}. The
$D9$'s have been reduced to $D5^\prime$'s by the use of T
dualization on the 4,5,8 and 9th coordinates, where there are two
sets of such states located at the origin.  This can be seen by
the integer and half integer massive states that couple to the
$D9$'s in the transverse channel. By performing T dualizations on
these coordinates the $D5$'s are effectively rotated to allow them
to wrap the $T_{89}$ torus. The fixed points relevant to the $D5$
branes are denoted by circles. The illustration thus shows a
rather standard geometry of the branes.  The lattice origin shows
the transverse annulus perfect squares as
\begin{eqnarray}
\tilde{{\cal A}}_o&=&\frac{2^{-5}}{8}v_1\bigg\{\big(N\sqrt{v_2v_3}+
\frac{2D}{\sqrt{v_2v_3}}\big)^2W_1Q_o+\big(N\sqrt{v_2v_3}-
\frac{2D}{\sqrt{v_2v_3}}\big)^2W_1Q_v\nn\\
&&+\big[N^2v_2v_3W_1(W_2W_3-1)+
\frac{4D^2}{v_2v_3}W_1(P^e_1P_3-1)\big](Q_o+Q_v)\bigg\}\nn\\
&&+\frac{2^{-5}}{4}v_1\bigg\{\big[(R_N-4R_D)^2+
7R_N^2\big]Q_s+\big[(R_N+4R_D)^2+7R_N^2\big]Q_c\bigg\}W_1\nn\\
&&\frac{2^{-5}}{4}v_1\times8R_N^2W_1^{n+\frac{1}{2}}(Q_s+Q_c).
\end{eqnarray}

The ambiguity in the coupling sign in the Mobius is chosen so as
to allow a consistent tadpole cancellation.  The transverse Mobius
is then provided as,
\begin{eqnarray}
\tilde{{\cal
M}}&=&-\frac{v_1}{4}\bigg\{\big[Nv_2v_3(W_1^eW_2^e+W_1^oW_2^e)W^e_3\nn\\
&&+\frac{2D}{v_2v_3}(W^e_1P^e_2+W^o_1(-1)^{m_2}P^e_2)P_3^e\big]
(\hat{Q}_o+\hat{Q}_v)\nn\\
&&+(NW_1+2DW^e_1)(\hat{Q}_o-\hat{Q}_v)
{\bigg(\frac{2\hat{\eta}}{\hat{\theta}_2}\bigg)}^2\bigg\}
\end{eqnarray}

The corresponding direct channel amplitudes for the annulus and
Mobius are
\begin{eqnarray}
{\cal A}&=&\frac{1}{8}\bigg\{\big[N^2\big(1+(-1)^{m_1+m_2}\big)P_1P_2P_3\nn\\
&&+2D^2P_1(W_2+W_2^{n+\frac{1}{2}})W_3\big](Q_o+Q_v)\nn\\
&&+4NDP_1(Q_s+Q_c){\bigg(\frac{\eta}{\theta_4}\bigg)}^2+
2\big[R_N^2P_1^e+R_D^2P_1\big](Q_o-Q_v){\bigg(\frac{2\eta}{\theta_2}\bigg)}^2\nn\\
&&+4R_NR_DP_1(Q_s-Q_c){\bigg(\frac{\eta}{\theta_3}\bigg)}^2\bigg\}
\end{eqnarray}and
\begin{eqnarray}
{\cal M}&=&-\frac{1}{8}\bigg\{\big[N\big(1+(-1)^{m_1+m_2}\big)P_1P_2P_3\nn\\
&&+2D\big(P_1W_2+(-1)^{m_1}P_1W_2^{n+\frac{1}{2}}\big)W_3\big](\hat{Q}_o+
\hat{Q}_v)\nn\\
&&-2(NP^e_1+DP_1)(\hat{Q}_o-\hat{Q}_v){\bigg(\frac{2\hat{\eta}}
{\hat{\theta}_2}\bigg)}^2\bigg\}
\end{eqnarray}

In order to extract the tadpole information attention must be paid
to the differing contributions from the $NS$ and R vacua.  The
Laurant modes
\begin{eqnarray}
L_n=\frac{1}{2}:\sum_n \alpha_{n-l}.\alpha_l:+\frac{1}{2}:\sum_w
(w-\frac{n}{2})\phi_{n-w}\phi_w:+\delta_{n,0}\Delta
\end{eqnarray}
for transverse oscillations acquire a total addition of
$-\frac{1}{16}(D-2)$ from the $NS$ sector and 0 from the R. The
zero modes in the amplitude $\tilde{{\cal K}}+\tilde{{\cal
A}}+\tilde{{\cal M}}$ give rise to divergent terms.  Such
contributions correspond to tadpole diagrams, which in a sense,
are vacuum bubbles terminating on branes or O-planes. Consistency
thus forces a constraint on the gauge group dimension which is
necessarily in the adjoint representation.  The construction so
far then yields the following tadpole conditions
\begin{eqnarray}
N=32,~2D=32,~R_N=0~and~R_D=0.\nn
\end{eqnarray}

The required Chan-Paton parameterization is then
\begin{eqnarray}
N=n+\bar{n},~D=d+\bar{d},~R_N=i(n-\bar{n})~and~R_D=i(d-\bar{d}).\nn
\end{eqnarray}
With these representations, one finds that the open sector has the
gauge group breaking
\begin{eqnarray}
U(16)_9\times U(16)_5\rightarrow U(16)_9\times U(8)_5\nn,
\end{eqnarray}
and shows the appropriate gauge couplings as
\begin{eqnarray}
{\cal A}_o+M_o&=&(n\bar{n}+d\bar{d})Q_o+\frac{1}{2}\big(n(n-1)
+\bar{n}(\bar{n}-1)+d(\bar{d}-1)+\bar{d}(\bar{d}-1)\big)Q_v\nn\\
&&+(n\bar{d}+\bar{n}d)Q_s+(nd+\bar{n}\bar{d})Q_c.
\end{eqnarray}
So, with the absence of sectors lying in the diagonal of modular
orbits, this model retains {\cal N}=2 supersymmetry in all
sectors. It has chiral matter in the form of hypermultiplets in
the representations $(120\oplus\overline{120},1)$,
$(1,28\oplus\overline{28})$ and $(16,\overline{8})$ under
$U(16)_9\times U(8)_5$.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% New Section
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{$T^{6}/\bbbb{Z}_{2}^3$ model}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% New Section
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

The projection that realizes the orbifold structure is represented
as
\begin{eqnarray}
\frac{1}{8}(1+g)(1+f)(1+\delta)\nn
\end{eqnarray}
The $\bb{Z}_2 \times \bb{Z}_2 \times \bb{Z}_2$ generators are
\begin{eqnarray}
&&g=(1,1;-1,-1;-1,-1),\nn\\&&f=(-1,-1;1,1;-1,-1),\nn\\
&&\delta=(A_1,1;A_1,1;A_1,1).\nn
\end{eqnarray}
Unlike the previous case, there are pure orbifold elements present
in the $\bb{Z}_2 \times \bb{Z}_2 \times \bb{Z}_2$ projection,
which will leave trace components on the diagonal of the matrix M
in (\ref{eqn:MatrixM}), so there will be terms in the torus
amplitude that are not determined by its modular invariance. These
terms will be realized in modular orbits as off diagonal twisted
sectors with orbifold insertions. As such, an ambiguity will be
present in the form of a sign freedom. This will give rise to
models with or without discrete torsion and necessitate the study
of different classes of models within a choice of sign as shown in
\cite{CAAS} for the $\bb{Z}_2 \times \bb{Z}_2$ case without
shifts. It will also create inconsistencies with the tadpole
conditions that arise from the $NS$ and $R$ sectors.

The torus amplitude results from projecting the type IIB trace as
\begin{eqnarray}
{\cal
T}=\frac{1}{8}TrP_{GSO}(1+g)(1+f)(1+\delta)q^{L_0}\bar{q}^{\tilde{L}_0}.\nn
\end{eqnarray}
with the explicit IIB projection factor of $\frac{1}{2}$ not shown
for brevity.

The models exhibit ${\cal N}=1$ SUSY which results from the
orbifold action on the Ramond sector so as to yield only one
independent fermionic ground state. The torus amplitude results
from the projected trace as
\begin{eqnarray}
{\cal T}&=&\frac{1}{8}\bigg{\{}|T_{oo}|^2\big{[}{\Lambda^1}_{m,n}
{\Lambda^2}_{m,n}{\Lambda^3}_{m,n}\nn\\
&&+{\Lambda^1}_{m,n+\frac{1}{2}}{\Lambda^2}_{m,n+\frac{1}{2}}
{\Lambda^3}_{m,n+\frac{1}{2}}\big{]}\big{(}1+(-1)^{m_1+m_2+m_3}\big{)}\nn\\
&&+|T_{ok}|^2{\Lambda^k}_{m,n}\big(1+(-1)^{m_k}\big){\vline\frac{2\eta}
{\theta_2}\vline}^4\nn\\
&&+16|T_{ko}|^2\big{(}{\Lambda^k}_{m,n}+{\Lambda^k}_{m,n+\frac{1}{2}}\big{)}
{\vline\frac{\eta}{\theta_4}\vline}^4 \nn\\
&&+16|T_{kk}|^2\big{(}{\Lambda^k}_{m,n}+(-1)^{m_k}
{\Lambda^k}_{m,n+\frac{1}{2}}\big{)}
{\vline\frac{\eta}{\theta_3}\vline}^4 \nn\\
&&+\epsilon(|T_{gh}|^2+|T_{gf}|^2+|T_{fg}|^2+|T_{fh}|^2+|T_{hg}|^2
+|T_{hf}|^2){\vline\frac{8{\eta}^3}
{\theta_2\theta_3\theta_4}\vline}^2\bigg{\}}.
\end{eqnarray}
For the terms $T_{km}$, which lie in the diagonal region of
orbifold/twisted sectors (or orbits outside of those that are
related by $S$ and $T$ transformations to the orbifold states), a
freedom to choose a sign is present. Discrete torsion is obtained
by taking $\epsilon=-1$, the resulting spectral content for cases
with and without torsion are different along with SUSY breaking in
different sectors by the possible presence of anti-branes.

The whole construction is done in the breaking from $SO(8)$ to
$SO(2)^4$ under $T^6/\bb{Z}_2\times \bb{Z}_2\times \bb{Z}_2 $
compactification. The following supersymmetric characters are
defined as:
\begin{eqnarray}\label{eqn:char}
\tau_{oo}&=&V_2O_2O_2O_2+O_2V_2V_2V_2-S_2S_2S_2S_2-C_2C_2C_2C_2\nn\\
\tau_{og}&=&O_2V_2O_2O_2+V_2O_2V_2V_2-C_2C_2S_2S_2-S_2S_2C_2C_2\nn\\
\tau_{oh}&=&O_2O_2O_2V_2+V_2V_2V_2O_2-C_2S_2S_2C_2-S_2C_2C_2S_2\nn\\
\tau_{of}&=&O_2O_2V_2O_2+V_2V_2O_2V_2-C_2S_2C_2S_2-S_2C_2S_2C_2\nn\\
\nn\\
\tau_{go}&=&V_2O_2S_2C_2+O_2V_2C_2S_2-S_2S_2V_2O_2-C_2C_2O_2V_2\nn\\
\tau_{gg}&=&O_2V_2S_2C_2+V_2O_2C_2S_2-S_2S_2O_2V_2-C_2C_2V_2O_2\nn\\
\tau_{gh}&=&O_2O_2S_2S_2+V_2V_2C_2C_2-C_2S_2V_2V_2-S_2C_2O_2O_2\nn\\
\tau_{gf}&=&O_2O_2C_2C_2+V_2V_2S_2S_2-S_2C_2V_2V_2-C_2S_2O_2O_2\nn\\
\nn\\
\tau_{ho}&=&V_2S_2C_2O_2+O_2C_2S_2V_2-C_2O_2V_2C_2-S_2V_2O_2S_2\nn\\
\tau_{hg}&=&O_2C_2C_2O_2+V_2S_2S_2V_2-C_2O_2O_2S_2-S_2V_2V_2C_2\nn\\
\tau_{hh}&=&O_2S_2C_2V_2+V_2C_2S_2O_2-S_2O_2V_2S_2-C_2V_2O_2C_2\nn\\
\tau_{hf}&=&O_2S_2S_2O_2+V_2C_2C_2V_2-C_2V_2V_2S_2-S_2O_2O_2C_2\nn\\
\nn\\
\tau_{fo}&=&V_2S_2O_2C_2+O_2C_2V_2S_2-S_2V_2S_2O_2-C_2O_2C_2V_2\nn\\
\tau_{fg}&=&O_2C_2O_2C_2+V_2S_2V_2S_2-C_2O_2S_2O_2-S_2V_2C_2V_2\nn\\
\tau_{fh}&=&O_2S_2O_2S_2+V_2C_2V_2C_2-C_2V_2S_2V_2-S_2O_2C_2O_2\nn\\
\tau_{ff}&=&O_2S_2V_2C_2+V_2C_2O_2S_2-C_2V_2C_2O_2-S_2O_2S_2V_2.\nn\\
\end{eqnarray}
Where one combines these into the character sums as
\begin{eqnarray}
&&T_{ko}=\tau_{ko}+\tau_{kg}+\tau_{kh}+\tau_{kf} \quad \quad
T_{kg}=\tau_{ko}+\tau_{kg}-\tau_{kh}-\tau_{kf}\nn\\ \nn\\
&&T_{kh}=\tau_{ko}-\tau_{kg}+\tau_{kh}-\tau_{kf} \quad \quad
T_{kf}=\tau_{ko}-\tau_{kg}-\tau_{kh}+\tau_{kf},
\end{eqnarray}
which for the sake of clarification, $k$ takes on the elements $o$
(the $\bb{Z}_2\times \bb{Z}_2$ identity) , $g$, $f$ and $h$.
Henceforth, $k$ will be confined to those elements which exclude
the identity. In addition, where ever a sum occurs in character
sets such as $T_{kl}$, it is taken that the condition $k \neq l$
applies.

All amplitudes are one loop expressions, as such it is easily seen
that the above separates into $NS-R$ sectors. The origin of the
torus is thus
\begin{eqnarray}
{\cal T}_0&=&\frac{1}{8}\bigg{\{}2|T_{oo}|^2+2|T_{ok}|^2
+16|T_{ko}|^2+16|T_{kk}|^2\bigg{\}}
\nn\\ \nn\\
&=&\frac{1}{8}\bigg{\{}8\big{(}|\tau_{oo}|^2+|\tau_{og}|^2
+|\tau_{of}|^2+|\tau_{oh}|^2\big{)}+64\big{(}\ldots\big{)}\bigg{\}}
\end{eqnarray}
which, as the previous $\bb{Z}_2\times \bb{Z}_2$ modulated torus
has 8 fixed points from each of the three twisted sectors, as
expected. As will be seen later, the shift action makes an
identification between fixed points thus halving the original
number of 16.  The Klein amplitude is given by
\begin{eqnarray}
{\cal
K}=\frac{1}{8}\bigg{\{}(P^{1}P^{2}P^{3}\big{(}1+(-1)^{m_1+m_2+m_3}\big{)}+
\big{(}1+(-1)^{m_1}\big{)}P^{1}W^{2}W^{3}\nn\\
+\big{(}1+(-1)^{m_2}\big{)}W^{1}P^{2}W^{3}+\big{(}1+(-1)^{m_3}\big{)}
W^{1}W^{2}P^{3})T_{oo}\nn\\
+2\times16\epsilon_k\bigg[P^k+
\epsilon\big(W^k+(-1)^{n_k}W^k_{n+\frac{1}{2}}\big)\bigg]\bigg{(}
\frac{\eta}{\theta_4}\bigg{)}^2T_{ko}\bigg{\}}
\end{eqnarray}
The closed sector amplitudes ${\cal T}+{\cal K}$ should give a
proper particle interpretation, such that it should contain 1
graviton at the lattice origin, and in an expansion of powers of
$|q|^2+q$, the combination should have integer coefficients.  The
later comprises the degeneracies for a given mass level. The
measure associated with the Klein for the parameter $\tau_2$ is
\begin{eqnarray}
\int\frac{d^2\tau}{{\tau_2}^3}\quad
\overrightarrow{t=2\tau_2}\quad2^2\int\frac{d^2t}{t^3}.
\end{eqnarray}
Poisson resummation gives a factor of 2 for each $T^2$ lattice
which is not acted on by an orbifold operation.  There is no
factorial contribution from orbifold lattices as it imposes the
condition of no momentum flow through the boundary. The resulting
transverse Klein amplitude is then
\begin{eqnarray}
\tilde{{\cal K}} =
\frac{2^5}{8}\bigg{\{}\big{(}v_1v_2v_2(W^1_eW^2_eW^3_e+W^1_oW^2_oW^3_o)
+\frac{v_k}{2v_lv_m}W^kP^l_eP^m_e\big{)}T_{oo}
\nn\\
\nn\\+2\epsilon_k\bigg[v_kW^k_e+\epsilon
\big(\frac{P^k_e}{v_k}+(-1)^{m_k}\frac{P^k_o}{v_k}\big)\bigg]
{\biggl(\frac{2\eta}{\theta_2}\biggr)}^2\tilde{T}_{ok}\bigg{\}}
\end{eqnarray}
The usual symmetrized summation convention is used for $k$,$l$ and
$m$. The transverse Klein amplitude at the origin is
\begin{eqnarray}
\tilde{{\cal K}}_o=\frac{2^5}{8}\bigg{\{}\big{(}v_1v_2v_2+\frac{v_k}
{2v_lv_m}\big{)}T_{oo}+2\epsilon_k\big(v_k+\epsilon\frac{1}{v_k}\big)
T_{ok}\bigg{\}}
\end{eqnarray}
%
%
which has an expanded form
%
%
\begin{eqnarray}
\tilde{{\cal K}_o}=\frac{2^5}{8}&&\bigg{\{}{\bigg{(}
\sqrt{v_1v_2v_3}+\epsilon_1\sqrt{\frac{v_1}{v_2v_3}}+
\epsilon_2\sqrt{\frac{v_2}{v_1v_3}}+
\epsilon_3\sqrt{\frac{v_3}{v_1v_2}}\bigg{)}}^2\tau_{oo}\nn\\
&&+{\bigg{(}\sqrt{v_1v_2v_3}+\epsilon_1\sqrt{\frac{v_1}{v_2v_3}}-
\epsilon_2\sqrt{\frac{v_2}{v_1v_3}}-
\epsilon_3\sqrt{\frac{v_3}{v_1v_2}}\bigg{)}}^2\tau_{og}\nn\\
&&+{\bigg{(}\sqrt{v_1v_2v_3}-\epsilon_1\sqrt{\frac{v_1}{v_2v_3}}+
\epsilon_2\sqrt{\frac{v_2}{v_1v_3}}-
\epsilon_3\sqrt{\frac{v_3}{v_1v_2}}\bigg{)}}^2\tau_{of}\nn\\
&&+{\bigg{(}\sqrt{v_1v_2v_3}-\epsilon_1\sqrt{\frac{v_1}{v_2v_3}}-
\epsilon_2\sqrt{\frac{v_2}{v_1v_3}}+
\epsilon_3\sqrt{\frac{v_3}{v_1v_2}}\bigg{)}}^2\tau_{oh}\bigg{\}}.\nn\\
\end{eqnarray}

The first difference in models with and without discrete torsion
is illustrated above by the brane content with different choices
of charges. Another difference will be the string spectral
content, which will yield alternative multiplet numbers and will
dictate the presence of particular multiplets, depending on the
choices of sign.

The annulus should contain $D9$ or (N) branes and $D5$ (D) branes.
Where in the transverse channel, the closed string propagating
between two N branes should have no momentum flow into or out of
the boundaries, and so $p_L=-p_R$ confines only winding modes to
be nonzero. Similarly for $D5$ branes which will have mixed
winding and momenta lattice modes.  The states that flow in the
torus must also flow in the annulus, so one must build on torus
states using corresponding $D5$ and $D9$ brane lattice terms. From
here, the supersymmetric character sets $T_{nm}$ separate into
combinations
\begin{eqnarray}\label{eqn:charsusybreak}
T_{nm}^{(\epsilon_i)}=T_{nm}^{NS}-\epsilon_i T_{nm}^{R}
\end{eqnarray}
where the choice of sign can signal brane SUSY breaking, by
introducing an antibrane for an appropriate sign of $\epsilon_k$.
To see this, (\ref{eqn:charsusybreak}) gives a fermionic
contribution that is not closed under $S$ transformation for
$\epsilon_k=-1$.  This then forces the characters to become
non-supersymmetric if one is to retain modular invariance without
changing the structure of the fermionic terms.  The characters
that are non-supersymmetric are defined as
\begin{eqnarray}
\tau_{oo}&=&O_2O_2O_2O_2+V_2V_2V_2V_2-C_2S_2S_2S_2-S_2C_2C_2C_2 \nn\\
\tau_{og}&=&V_2V_2O_2O_2+O_2O_2V_2V_2-S_2C_2S_2S_2-C_2S_2C_2C_2 \nn\\
\tau_{oh}&=&V_2O_2O_2V_2+O_2V_2V_2O_2-S_2S_2S_2C_2-C_2C_2C_2S_2 \nn\\
\tau_{of}&=&V_2O_2V_2O_2+O_2V_2O_2V_2-S_2S_2C_2S_2-C_2C_2S_2C_2 \nn\\
\nn\\
\tau_{go}&=&O_2O_2S_2C_2+V_2V_2C_2S_2-C_2S_2V_2O_2-S_2C_2O_2V_2 \nn\\
\tau_{gg}&=&V_2V_2S_2C_2+O_2O_2C_2S_2-C_2S_2O_2V_2-S_2C_2V_2O_2 \nn\\
\tau_{gh}&=&V_2O_2S_2S_2+O_2V_2C_2C_2-S_2S_2V_2V_2-C_2C_2O_2O_2 \nn\\
\tau_{gf}&=&V_2O_2C_2C_2+O_2V_2S_2S_2-C_2C_2V_2V_2-S_2S_2O_2O_2 \nn\\
\nn\\
\tau_{ho}&=&O_2S_2C_2O_2+V_2C_2S_2V_2-S_2O_2V_2C_2-C_2V_2O_2S_2 \nn\\
\tau_{hg}&=&V_2C_2C_2O_2+O_2S_2S_2V_2-S_2O_2O_2S_2-C_2V_2V_2C_2 \nn\\
\tau_{hh}&=&V_2S_2C_2V_2+O_2C_2S_2O_2-C_2O_2V_2S_2-S_2V_2O_2C_2 \nn\\
\tau_{hf}&=&V_2S_2S_2O_2+O_2C_2C_2V_2-S_2V_2V_2S_2-C_2O_2O_2C_2 \nn\\
\nn\\
\tau_{fo}&=&O_2S_2O_2C_2+V_2C_2V_2S_2-C_2V_2S_2O_2-S_2O_2C_2V_2 \nn\\
\tau_{fg}&=&V_2C_2O_2C_2+O_2S_2V_2S_2-S_2O_2S_2O_2-C_2V_2C_2V_2 \nn\\
\tau_{fh}&=&V_2S_2O_2S_2+O_2C_2V_2C_2-S_2V_2S_2V_2-C_2O_2C_2O_2 \nn\\
\tau_{ff}&=&V_2S_2V_2C_2+O_2C_2O_2S_2-S_2V_2C_2O_2-C_2O_2S_2V_2. \nn\\
\end{eqnarray}\label{eqn:NonSUSYChar}
The transverse annulus amplitude with undetermined factors, which
will be determined further on, has the form
\begin{eqnarray}
\tilde{{\cal A}}=&\frac{2^{-5}}{8}&\bigg{\{}\bigg{(}2M_1N_o^2v_1v_2v_3
(W^1W^2W^3+W^1_{n+\frac{1}{2}}W^2_{n+\frac{1}{2}}W^3_{n+\frac{1}{2}})\nn\\
&&+M_2\frac{v_k}{2v_lv_m}D^2_{k;o}W^kP^lP^m\big{(}1
+(-1)^{m_l+m_m}\big{)}\bigg{)}T_{oo}\nn\\
&&+\bigg[M_3N_k^2v_k(W^k+W^k_{n+\frac{1}{2}})\nn\\
&&+M_4D_{k;k}^2v_kW^k+M_5D^2_{l;k}
\frac{P^k}{v_k}\bigg]\tilde{T}_{ko}{\bigg{(}\frac{\eta}
{\theta_4}\bigg{)}}^2\nn\\
&&+2M_6N_oD_{k;o}v_kW^k\tilde{T}_{ok}^{(\epsilon_k)}{\bigg{(}
\frac{2\eta}{\theta_2}\bigg{)}}^2\nn\\
&&+M_7N_kD_{k;k}v_kW^k\tilde{T}_{kk}^{(\epsilon_k)}{\bigg{(}
\frac{\eta}{\theta_3}\bigg{)}}^2\nn\\
&&+M_8N_lD_{k;l}\tilde{T}_{lk}^{(\epsilon_k)}\frac{8{\eta}^3}
{\theta_2\theta_3\theta_4}\nn\\
&&+M_9D_{k;o}D_{l;o}\frac{P^m}{v_m}\big{(}1+(-1)^{m_m}\big{)}
\tilde{T}_{om}^{(\epsilon_k\epsilon_l)}{\bigg{(}\frac{2\eta}
{\theta_2}\bigg{)}}^2\nn\\
&&+M_{10}D_{k;m}D_{l;m}\frac{P^m}{v_m}\tilde{T}_{mm}^{(\epsilon_k
\epsilon_l)}{\bigg{(}\frac{\eta}{\theta_3}\bigg{)}}^2\nn\\
&&+M_{11}D_{k;k}D_{l;k}\tilde{T}_{km}^{(\epsilon_k\epsilon_l)}
\frac{8{\eta}^3}{\theta_2\theta_3\theta_4}\bigg{\}}.
\end{eqnarray}
The various brane - lattice couplings are defined in table
(\ref{tab:cc}).
\begin{table}
\begin{center}
\begin{tabular}{|c|c|}\hline
Plane diagrams & Volumes \\ \hline $D9-D9$, $D9-O9$,
$O9-O9$ & $v_1v_2v_3$ \\
$D5_k-D5_k$, $D5_k-O5_k$, $O5_k-O5_k$ & $\frac{v_k}{v_lv_m}$ \\
$D5_k-D5_l$, $D5_k-O5_l$, $O5_k-O5_l$ & $\frac{1}{v_m}$ \\
$D9-D5_k$, $D9-O5_k$, $D5_k-O9$ & $v_k$ \\ \hline \hline
$\tilde{\cal A}$ and $\tilde{\cal K}$ Plane Diagrams & Lattice
Couplings
\\\hline
$D9-D9$ & $W^1W^2W^3+W^1_{n+\frac{1}{2}}W^2_{n+\frac{1}{2}}
W^3_{n+\frac{1}{2}}$
\\
$D5_k-D5_k$ & $W^kP^lP^m\big(1+(-1)^{m_l+m_m}\big)$ \\
$O9-O9$ & $W^1_eW^2_eW^3_e+W^1_oW^2_oW^3_o$ \\
$O5_k-O5_k$ & $W^kP^l_eP^m_e$ \\ \hline \hline $\tilde{\cal M}$
plane diagrams & Lattice Couplings
\\\hline
$D9-O9$ & $W^1_eW^2_eW^3_e+W^1_oW^2_oW^3_o$ \\
$D9-O5_k$ & $W^k$ \\
$D5_k-O9$ & $W^k_e$ \\
$D5_k-O5_k$ & $W^k_eP^l_eP^m_e+(-1)^{m_l+m_m}W^k_oP^l_eP^m_e$ \\
$D5_k-O5_l$ & $P^m_e$ \\\hline
\end{tabular}
\end{center}
\caption{Lattice restrictions}\label{tab:cc}
\end{table}

One must now fix the relative factors for the different diagrams
by looking at the various sectors and arranging them into perfect
squares. The zero mode contribution of $\tilde{A}$ is given by
\begin{eqnarray}
\tilde{{\cal A}}_o=&\frac{2^{-5}}{8}&\bigg{\{}
\bigg{(}2M_1N_o^2v_1v_2v_3+2M_2\frac{v_k}{2v_lv_m}D^2_{k;o}\bigg{)}
T_{oo}\nn\\ \nn\\
&&+\bigg[(M_3N_k^2+M_4D_{k;k}^2)v_k+M_5D^2_{l;k}\frac{1}{v_k}\bigg]
\tilde{T}_{ko}+2M_6N_oD_{k;o}v_k\tilde{T}_{ok}^{(\epsilon_k)}\nn\\ \nn\\
&&+M_7N_kD_{k;k}v_k\tilde{T}_{kk}^{(\epsilon_k)}+4M_8N_lD_{k;l}
\tilde{T}_{lk}^{(\epsilon_k)}\nn\\ \nn\\
&&+2M_9D_{k;o}D_{l;o}\frac{1}{v_m}\tilde{T}_{om}^{(\epsilon_k\epsilon_l)}+
M_{10}D_{k;m}D_{l;m}\frac{1}{v_m}
\tilde{T}_{mm}^{(\epsilon_k\epsilon_l)}\nn\\ \nn\\
&&+4M_{11}D_{k;k}D_{l;k}\tilde{T}_{kk}^{(\epsilon_k\epsilon_l)}\bigg{\}}.
\end{eqnarray}
{}From this, one can read off the untwisted zero lattice mode
contribution as
\begin{eqnarray}\label{eqn:untwnu}
\tilde{{\cal A}}_{o}&=&\frac{2^{-5}}{8}\bigg{\{}\bigg{(}
2M_1N_o^2v_1v_2v_3+2M_2\frac{v_k}{2v_lv_m}D^2_{k;o}\bigg{)}T_{oo}\nn\\ \nn\\
&&+2M_6N_oD_{k;o}v_k\tilde{T}_{ok}^{(\epsilon_k)}+2M_9D_{k;o}D_{l;o}
\frac{1}{v_m}\tilde{T}_{om}^{(\epsilon_k \epsilon_l)}\bigg\}\nn\\
\end{eqnarray}
which has a perfect square arrangement
\begin{eqnarray}\label{eqn:AnnulusOrigin}
\tilde{{\cal A}}_{o}&&=\frac{2^{-5}}{8}\bigg{\{}{\bigg{(}
N_o\sqrt{v_1v_2v_3}+D_{g;o}\sqrt{\frac{v_1}{v_2v_3}}+D_{f;o}
\sqrt{\frac{v_2}{v_lv_3}}+D_{h;o}\sqrt{\frac{v_3}{v_1v_2}}\bigg{)}}^2
\tau_{oo}^{NS}\nn\\ \nn\\
&&-{\bigg{(}N_o\sqrt{v_1v_2v_3}+\epsilon_1D_{g;o}
\sqrt{\frac{v_1}{v_2v_3}}+\epsilon_2D_{f;o}
\sqrt{\frac{v_2}{v_1v_3}}+\epsilon_3D_{h;o}\sqrt{\frac{v_3}{v_1v_2}}
\bigg{)}}^2\tau_{oo}^{R}\nn\\ \nn\\
&&+{\bigg{(}N_o\sqrt{v_1v_2v_3}+D_{g;o}\sqrt{\frac{v_1}{v_2v_3}}-D_{f;o}
\sqrt{\frac{v_2}{v_1v_3}}-D_{h;o}\sqrt{\frac{v_3}{v_1v_2}}\bigg{)}}^2
\tau_{og}^{NS}\nn\\ \nn\\
&&-{\bigg{(}N_o\sqrt{v_1v_2v_3}+\epsilon_1D_{g;o}\sqrt{\frac{v_1}{v_2v_3}}
-\epsilon_2D_{f;o}\sqrt{\frac{v_2}{v_1v_3}}-\epsilon_3D_{h;o}
\sqrt{\frac{v_3}{v_1v_2}}\bigg{)}}^2\tau_{og}^{R}\nn\\ \nn\\
&&+{\bigg{(}N_o\sqrt{v_1v_2v_3}-D_{g;o}\sqrt{\frac{v_1}{v_2v_3}}+D_{f;o}
\sqrt{\frac{v_2}{v_1v_3}}-D_{h;o}\sqrt{\frac{v_3}{v_1v_2}}\bigg{)}}^2
\tau_{of}^{NS}\nn\\ \nn\\
&&-{\bigg{(}N_o\sqrt{v_1v_2v_3}-\epsilon_1D_{g;o}\sqrt{\frac{v_1}{v_2v_3}}+
\epsilon_2D_{f;o}\sqrt{\frac{v_2}{v_1v_3}}-\epsilon_3D_{h;o}
\sqrt{\frac{v_3}{v_1v_2}}\bigg{)}}^2\tau_{of}^{R}\nn\\ \nn\\
&&+{\bigg{(}N_o\sqrt{v_1v_2v_3}-D_{g;o}\sqrt{\frac{v_1}{v_2v_3}}-
D_{f;o}\sqrt{\frac{v_2}{v_1v_3}}+D_{h;o}\sqrt{\frac{v_3}{v_1v_2}}
\bigg{)}}^2\tau_{oh}^{NS}\nn\\ \nn\\
&&-{\bigg{(}N_o\sqrt{v_1v_2v_3}-\epsilon_1D_{g;o}
\sqrt{\frac{v_1}{v_2v_3}}-\epsilon_2D_{f;o}\sqrt{\frac{v_2}{v_1v_3}}
+\epsilon_3D_{h;o}\sqrt{\frac{v_3}{v_1v_2}}\bigg{)}}^2
\tau_{oh}^{R}\bigg{\}}.\nn\\
\end{eqnarray}
The factors in (\ref{eqn:untwnu}) are then fixed by this necessary
form as $M_1=M_2=M_9=\frac{1}{2}$ and $M_6=1$).

For the twisted sectors, one has to separate a series of terms,
the coefficients of which must add up to the total number of fixed
points.  The coefficients of the individual terms are derived as
follows. If you take $N_g$ term, this fills all compact and
non-compact dimensions.  It is therefore wrapped around all
compact dimensions and sees all the fixed points. The coefficient
formula is
\begin{eqnarray}\label{eqn:FixedPointSummary}
\sqrt{\frac{{\rm number~of~fixed~points}}{{\rm
number~of~seen~fixed~points}}}
\end{eqnarray}
The same applies to the $v_k$ characters.  So in the $N_g$ case,
it wraps around all tori, so in the g-twisted sector it is not
localized in the first tori.  $N_g$ thus has the coefficient
\begin{eqnarray}
\sqrt{\frac{16}{16}}\sqrt{v_k}
\end{eqnarray}

When considering the charges involved with terms like $D_{k;l}$,
they are built by understanding that $l$ represents the fixed
point configuration of $T^2_{45} \times T^2_{67} \times T^2_{89}$
and $k$ represents whether the brane is wrapped or transverse. For
example, $D_{g;f}$ has fixed points in the first and third torus
corresponding to $f$. The generator $g$ implies that $D_{g;f}$ is
wrapped around the first tori and is transverse to the second and
third, hence, it \textit{sees} four fixed points.  Looking firstly
at the g-twisted sector. This initially has 16 fixed points
located in the second and third tori which under the operation of
the shift, are reduced to 8. So, the remaining independent fixed
points are as in table \ref{tab:shiftedFixedPoints}.  One now
needs to fix the coefficients of the origin terms which give rise
to the twisted perfect square contributions.  For the $g$-twisted
sector, one has terms in the annulus as
\begin{eqnarray}
\tilde{{\cal A}}^g=&\frac{2^{-5}}{8}&\bigg{\{}
\bigg[(M_3N_g^2+M_4D_{g;g}^2)v_1+M_5D^2_{l;g}\frac{1}{v_1}\bigg]
\tilde{T}_{go}\nn\\ \nn\\
&&+M_7N_gD_{g;g}v_1\tilde{T}_{gg}^{(\epsilon_1)}\nn\\ \nn\\
&&+4M_8N_gD_{k;g}\tilde{T}_{gk}^{(\epsilon_k)}\nn\\ \nn\\
&&+M_{10}D_{k;g}D_{l;g}\frac{1}{v_1}
\tilde{T}_{gg}^{(\epsilon_k\epsilon_l)}\nn\\ \nn\\
&&+4M_{11}D_{g;g}D_{l;g}\tilde{T}_{gm}^{(\epsilon_k\epsilon_l)}\bigg\}.\nn\\
\end{eqnarray}

Now, under the identification of the fixed points, one can
categorize the types of brane that see certain fixed points. Then
by taking brane occupation of both identified points, for example
\begin{eqnarray}
(0,0;0,\frac{1}{2})\sim(\frac{1}{2},0;,\frac{1}{2},\frac{1}{2})\nn\\
\end{eqnarray}
requires an expression involving $N_g$, $D_{h;g}$ and $N_g$.  The
types of $D5$ brane involved with associated fixed points arise
due to an orbifold twist in a torus direction making a brane
transverse to it, whereas the absence of such an operation implies
the brane wraps that direction.  The resulting perfect square
structure for the $\tau_{gl}$ with orbifold element $g$ is
\begin{eqnarray}
\tilde{{\cal A}}^g_{o}&=&2\times\frac{2^{-5}}{8}\bigg{\{}{\big{(}
\sqrt{v_1}N_g+4s_1\sqrt{v_1}D_{g;g}+2s_2\frac{1}{\sqrt{v_1}}D_{f;g}
+2s_3\frac{1}{\sqrt{v_1}}D_{h;g}\big{)}}^2\nn\\ \nn\\
&&+3{\big{(}\sqrt{v_1}N_g+2s_4\frac{1}{\sqrt{v_1}}D_{f;g}\big{)}}^2
+3{\big{(}\sqrt{v_1}N_g+2s_5\frac{1}{\sqrt{v_1}}D_{h;g}\big{)}}^2
+v_1N_g^2\bigg{\}},\nn\\
\end{eqnarray}
where the signs $s_i$ are completely determined by the orbifold
direction $o,g,f$ and $h$ within the $g$ twist and the sector that
is considered, $NS$ or $R$.  The overall factor of 2 is to account
for the multiplicity of the shifted fixed points.  The first
square term represents the brane occupation of the $(0,0;0,0)$
fixed point of the second and third torus, the second square term
corresponds to $(0,0;\frac{1}{2},0)$, $(0,0;0,\frac{1}{2})$ and
$(0,0;\frac{1}{2},\frac{1}{2})$ fixed points, and so has
multiplicity $3$. This is similarly achieved for the other twisted
sectors. Here it is easily seen that the sum of the perfect square
coefficients should reflect the total number of fixed points.  The
charge factors of the various branes can be seen by using the
prescription of (\ref{eqn:FixedPointSummary}), for example, the
factors in the second term arise as \beqn
\sqrt{\frac{4\times4}{4\times1}},
\sqrt{\frac{4\times4}{1\times1}}.\nn \eeqn

With the above structure of the twisted square contributions, one
can fix the remaining factors $M_i$.  With the aid of the identity
\begin{eqnarray}
\theta_2 \theta_3 \theta_4 = 2\eta^3,\nn
\end{eqnarray}
the twisted factors can be identified.  The transverse annulus is
now
\begin{eqnarray}
\tilde{{\cal A}}=&\frac{2^{-5}}{8}&\bigg{\{}\bigg{(}
N_o^2v_1v_2v_3(W^1W^2W^3+W^1_{n+\frac{1}{2}}W^2_{n+\frac{1}{2}}
W^3_{n+\frac{1}{2}})\nn\\
&&+\frac{v_k}{2v_lv_s}D^2_{k;o}W^kP^lP^s\frac{\big{(}1
+(-1)^{m_l+m_s}\big{)}}{2}\bigg{)}T_{oo}\nn\\
&&+2\times2\bigg{[}N_k^2v_k(W^k+W^k_{n+\frac{1}{2}})\nn\\
&&+2D_{k;k}^2v_kW^k+2D^2_{l;k}\frac{P^k}{v_k}\bigg{]}
\tilde{T}_{ko}{\bigg{(}\frac{2\eta}{\theta_4}\bigg{)}}^2\nn\\
&&+2N_oD_{k;o}v_kW^k\tilde{T}_{ok}^{(\epsilon_k)}{\bigg{(}
\frac{2\eta}{\theta_2}\bigg{)}}^2\nn\\
&&+2\times2N_kD_{k;k}v_kW_k\tilde{T}_{kk}^{(\epsilon_k)}{\bigg{(}
\frac{2\eta}{\theta_3}\bigg{)}}^2\nn\\
&&+2\times4N_lD_{k;l}\tilde{T}_{lk}^{(\epsilon_k)}\frac{8{\eta}^3}
{\theta_2\theta_3\theta_4}\nn\\
&&+D_{k;o}D_{l;o}\frac{P^s}{v_s}\frac{\big{(}1+(-1)^{m_s}\big{)}}{2}
\tilde{T}_{os}^{(\epsilon_k\epsilon_l)}{\bigg{(}\frac{2\eta}
{\theta_2}\bigg{)}}^2\nn\\
&&+2D_{k;m}D_{l;m}\frac{P^m}{v_m}\tilde{T}_{mm}^{(\epsilon_k\epsilon_l)}
{\bigg{(}\frac{2\eta}{\theta_3}\bigg{)}}^2\nn\\
&&+2\times4D_{k;k}D_{l;k}\tilde{T}_{km}^{(\epsilon_k\epsilon_l)}
\frac{8{\eta}^3}{\theta_2\theta_3\theta_4}\bigg{\}}.\nn\\
\end{eqnarray}

The relative placements of the $D5$'s is shown in figure
\ref{aZ2xZ2xZ21}.  The diagram shows $D5_{g;o}$, $D5_{f;o}$ and
$D5_{h;o}$ distinct processes within the annulus amplitude
according to the coordinates they wrap, which can be seen in the
amplitude as the correspondences of the wrapping $D_{g;o}\sim$45,
$D_{f;o}\sim$67 and $D_{h;o}\sim$89.  The effect of the diagonal
geometry of the $D5$ branes is a feature of freely acting shifts,
and in particular, the cases we consider when the shift is an
additional modulating group outside that of the orbifold
$\bb{Z}_2\times \bb{Z}_2$.

%%%%%%%%%%%%%%%%%%%%%%Transverse Annulus D5 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}
\centerline{\epsfxsize 2.0 truein \epsfbox {aZ2xZ2xZ21.eps}}
\caption{$D5_{k;o}$ configurations} \label{aZ2xZ2xZ21}
\end{figure}
%%%%%%%%%%%%%%%%%%%%% END OF FIGURE %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

The direct channel is found to be
\begin{eqnarray}\label{eqn:dirannorigin}
{\cal A}=&\frac{1}{8}&\bigg{\{}\bigg{(}N_o^2P^1P^2P^3\big(1
+(-1)^{m_1+m_2+m_3}\big)\nn\\
&&+\frac{1}{2}\frac{D^2_{k;o}}{2}P^k(W^lW^m+W^l_{n+\frac{1}{2}}W^m_{n
+\frac{1}{2}}\bigg{)}T_{oo}\nn\\
&&+\bigg{[}N_k^2P^k\big(1+(-1)^{m_k}\big)\nn\\
&&+2D_{k;k}^2P^k+2D^2_{l;k}W^k\bigg{]}T_{ok}{\bigg{(}
\frac{2\eta}{\theta_2}\bigg{)}}^2\nn\\
&&+2N_oD_{k;o}P^k
T_{ko}^{(\epsilon_k)}{\bigg{(}\frac{\eta}{\theta_4}\bigg{)}}^2\nn\\
&&-2\times2N_kD_{k;k}P^kT_{kk}^{(\epsilon_k)}{\bigg{(}\frac{\eta}
{\theta_3}\bigg{)}}^2\nn\\
&&+2\times2i(-1)^{k+l}N_lD_{k;l}T_{kl}^{(\epsilon_k)}\frac{2{\eta}^3}
{\theta_2\theta_3\theta_4}\nn\\
&&+\frac{1}{2}D_{k;o}D_{l;o}(W^m+W^m_{n+\frac{1}{2}})
T_{mo}^{(\epsilon_k\epsilon_l)}{\bigg{(}\frac{\eta}{\theta_4}\bigg{)}}^2\nn\\
&&-2\times
D_{k;m}D_{l;m}W^mT_{mm}^{(\epsilon_k\epsilon_l)}{\bigg{(}\frac{\eta}
{\theta_3}\bigg{)}}^2\nn\\
&&+2\times2i(-1)^{m+k}D_{k;k}D_{l;k}T_{mk}^{(\epsilon_k\epsilon_l)}
\frac{2{\eta}^3}{\theta_2\theta_3\theta_4}\bigg{\}}.\nn\\
\end{eqnarray}
Where it can be seen from (\ref{eqn:char}) that the following
character sets transform in the following manner under $S$: \beq
T_{mm}\rightarrow-T_{mm}\quad{\rm and}\quad T_{kl}\rightarrow
i(-1)^{k+l}T_{kl}. \eeq The others are invariant.  The values
$k$,$m$ and $l$ are $1,2$ and $3$ in correspondence with the
generators $g,f$ and $h$ respectively.  This can be seen simply by acting with
the operator $S$ (\ref{eqn:tran}) on the characters (\ref{eqn:char}), which has
the form
\begin{eqnarray}
S_{2n}=\frac{1}{2}\left(\matrix{ 1 & 1 & 1 & 1 \cr 1 & 1& -1 & -1
\cr 1 & -1 & i^{-n} & -i^{-n} \cr 1 & -1 & -i^{-n} & i^{-n} \cr
}\right),
\end{eqnarray}
and acts on the transverse of the vector
$(O_{2n},V_{2n},S_{2n},C_{2n})$ for an $SO(2n)$ group.

In constructing the Mobius it will be necessary to perform $P$
transforms on the amplitude components in order to gain the direct
channel equation.  Formally, the $P$ operator is a combination of
the already understood $S$ and $T$ transforms as
\begin{eqnarray}
P=TST^2S\nn
\end{eqnarray}
and so acts on the measure as
\begin{eqnarray}\label{eqn:mobmeas}
P:\frac{1}{2}+i\frac{\tau_2}{2}\quad\rightarrow \quad
\frac{1}{2}+\frac{i}{2{\tau_2}}
\end{eqnarray}
Combinations of the $S$ and $T$ operators satisfy
\begin{eqnarray}
S^2=(ST)^3=C\quad\Rightarrow\quad P^2=C
\end{eqnarray}
where $C$ is the charge conjugation matrix, which in these cases,
is simply the identity. It is then easy to see that this operation
acts on the Mobius lattice modes as in table \ref{tab:MobLatTrans}
\begin{table}[!ht]
\begin{center}
\begin{tabular}{|rllrll|}\hline
$P_m$ & $\rightarrow$ & $W_{2n}$ & $P_{2m}$ & $\rightarrow$ & $W_{n}$ \\
$(-1)^mP_m$ & $\rightarrow$ & $W_{2n+1}$ &
$(-1)^mP_{2m}$ & $\rightarrow$ & $W_{n+\frac{1}{2}}$\\ \hline
\end{tabular}
\end{center}
\caption{Mobius lattice transforms}\label{tab:MobLatTrans}
\end{table}
which shows this as just as an $S$-transformation would act on the
Klein states. This then implies an action on the characters as
\begin{eqnarray}\label{eqn:Pmatrix}
P=\left(\matrix{ c & s & 0 & 0 \cr s & -c & 0 & 0 \cr 0 & 0 & \chi
c & i\chi s \cr 0 & 0 & i\chi s & \chi c }\right)
\end{eqnarray}
for $s=sin(n\pi/4)$, $c=cos(n\pi/4)$ and
$\chi=e^{-i\frac{n\pi}{4}}$, for an $SO(2n)$ breaking.

Having fixed the relevant factors in the annulus, the Mobius can
now be constructed. The Mobius should symmetrize the Klein and the
annulus in the transverse channel, and must give a proper particle
interpretation with the annulus in the direct channel. Since the
Mobius in the transverse channel is a closed string propagating
between a brane and an $O$ plane, it is necessary to understand the
constraints on the Kaluza Klein and winding terms placed by the
brane and $O$-plane diagrams.  Table \ref{tab:cc} shows the
constrained lattice terms as read off of $\tilde{{\cal A}}$ and
$\tilde{{\cal K}}$. Taking the cross couplings of the various
$\tilde{\tau}_{ok}$ from the origin states of $\tilde{{\cal A}}_o$
and $\tilde{{\cal K}}_o$, the Mobius origin reads
\begin{eqnarray}\label{eqn:mobzm}
\tilde{{\cal M}}_o&=&\pm\frac{1}{4}\bigg{\{}N_ov_1v_2v_3\hat{T}_{oo}+\epsilon_k
D_{k;o}\frac{v_k}{2v_lv_m}\hat{\tilde{T}}_{oo}^{(\epsilon_k)}\nn\\
&&+\epsilon_kN_ov_k\hat{\tilde{T}}_{ok}+D_{k;o}v_k
\hat{\tilde{T}}_{ok}^{(\epsilon_k)}+\epsilon_mD_{l;o}\frac{1}{v_k}
\hat{\tilde{T}}_{ok}^{(\epsilon_l)}\bigg{\}}
\end{eqnarray}
The hatted characters signify the Mobius measure as defined in
(\ref{eqn:mobmeas}). There is a sign ambiguity from the square
root of various coupling terms, for example, the $N_o^2$ will have
Mobius coupling
\begin{eqnarray}
\tilde{\cal
M}_{O9-D9}^c=\pm2\times\sqrt{\frac{2^5}{8}}\times\sqrt{\frac{2^{-5}N_o^2}{8}}
\end{eqnarray}
where there is a diagram symmetry factor of $2$.  From here, the
massive modes must be built by taking $\tilde{A}_o$ and
$\tilde{{\cal K}}_o$ common massive modes.  In the direct annulus,
there are only integer lattice modes present on the $D9$-$D9$
coupling, so the resulting term is thus
\begin{eqnarray}
\tilde{\cal M}_{O9-D9}=\pm\frac{N_o}{4}v_1v_2v_3
(W^1_eW^2_eW^3_e+W^1_oW^2_oW^3_o)\hat{T}_{oo}
\end{eqnarray}
The $D_{k;o}D_{k;o}$ diagram is a little more subtle as the direct
annulus also contains $\frac{1}{2}$ integer lattice modes.  One
must now leave the transverse open process symmetrization
undisturbed while building in compatibility with the $\frac{1}{2}$
integer terms in the direct channel so as to achieve the desired
direct channel particle symmetrization of common modes.  So
\begin{eqnarray}
\tilde{\cal M}_{D5_k-O5_k}=\pm\frac{1}{4}\epsilon_kD_{k;o}
\frac{v_k}{2v_lv_s}\big(W^k_eP^l_eP^m_e+(-1)^{m_l+m_s}W^k_oP^l_eP^s_e\big)
\end{eqnarray}
This can easily be seen to symmetrize properly with the
corresponding annulus term by $P$ transforming appropriately. With
the inclusion of the orbifold sector, which now exhausts the
Mobius terms due to the form of the Klein, one finds
\begin{eqnarray}
\tilde{{\cal M}}&=&-\frac{1}{4}\bigg{\{}N_ov_1v_2v_3(W^1_eW^2_eW^3_e
+W^1_oW^2_oW^3_o)\hat{T}_{oo}\nn\\
&&+\epsilon_kD_{k;o}\frac{v_k}{2v_lv_s}\big(W^k_eP^l_eP^s_e
+(-1)^{m_l+m_s}W^k_oP^l_eP^s_e\big)\hat{\tilde{T}}_{oo}^{(\epsilon_k)}\nn\\
&&+\big(\epsilon_kN_ov_kW^k\hat{\tilde{T}}_{ok}+D_{k;o}v_kW^k_e
\hat{\tilde{T}}_{ok}^{(\epsilon_k)}\big){\bigg(\frac{2\hat{\eta}}
{\hat{\theta}_2}\bigg)}^2\nn\\
&&+\epsilon_mD_{l;o}\frac{P^k_e}{v_k}\hat{\tilde{T}}_{ok}^{(\epsilon_l)}
{\bigg(\frac{2\hat{\eta}}{\hat{\theta}_2}\bigg)}^2\bigg{\}}
\end{eqnarray}
Here, the $N_o-O$-plane coupling that resides in the orbifold
sector has both even and odd windings, this is to ensure correct
symmetrization with the direct annulus.

The Mobius origin (\ref{eqn:mobzm}) yields different charges for
the brane-$O$-plane couplings. The sign ambiguity from the
coupling constants is restricted to $(-)$ in all diagrams, this
will be apparent later as tadpole consistency condition.

The corresponding direct channel is obtained by $P$
transformation. It is noted that while $P$ has non trivial effect
of the lattice modes, it leaves the characters unchanged with the
exception of a sign change for the orbifold sector. This can be
seen by the representation defined in (\ref{eqn:Pmatrix}).  As
such
\begin{eqnarray}
{\cal M}&=&-\frac{1}{8}\bigg{\{}N_oP^1P^2P^3\big(1+(-1)^{m_1+m_2+m_3}\big)
\hat{T}_{oo}\nn\\
&&+\epsilon_k\frac{1}{2}D_{k;o}\big(P^kW^lW^m+(-1)^{m_k}P^kW^l_{n
+\frac{1}{2}}W^m_{n+\frac{1}{2}}\big)\hat{T}_{oo}^{(\epsilon_k)}\nn\\
&&-\big(2\epsilon_kN_oP^k_e\hat{T}_{ok}+D_{k;o}P^k
\hat{T}_{ok}^{(\epsilon_k)}\big){\bigg(\frac{2\hat{\eta}}
{\hat{\theta}_2}\bigg)}^2\nn\\
&&-\epsilon_mD_{l;o}W^k\hat{T}_{ok}^{(\epsilon_l)}{\bigg(
\frac{2\hat{\eta}}{\hat{\theta}_2}\bigg)}^2\bigg{\}}
\end{eqnarray}

By virtue of (\ref{eqn:charsusybreak}), it would seem that there
are tachyonic modes present here with the presence of
$V_2O_2O_2O_2\rightarrow O_2O_2O_2O_2$ (under the requirement that
the parent character $\tau_{oo}$ be closed under $S$). However,
the $P$ transformation which is structured differently form the
usual $S$ transformation, maintains a closed form of the
characters with $\epsilon_k=-1$. The terms in the direct channel
Mobius amplitude are thus free of tachyonic states. The direct
annulus has no terms of the form $T_{ok}^{(-1)}$ or
$T_{oo}^{(-1)}$, and so the model is tachyon free.  This is the
case for the parent $\bb{Z}_2\times \bb{Z}_2$ model, and thus is
true for any further modulation of it.

Having constructed all the necessary diagrams, a proper particle
interpretation must be forthcoming. The sum of the open and closed
diagrams must realize one graviton in the spectrum by looking at
the closed sector only, one can see that this is indeed the case.
Also, in the open sector, an interpretation is only valid in the
case that the gauge group for the $D5$ branes is halved.  This is
a generic artifact of making a fixed point identification as has
been used.

The tadpole conditions are
\begin{eqnarray}
\frac{2^5}{8}+\frac{2^{-5}}{8}N_o^2-\frac{N_o}{4}=0~\Rightarrow~N_o=32.\nn
\end{eqnarray}
It is seen that if discrete torsion is present, the tadpole
conditions in the $NS$ and $R$ sector fail to be consistent.  The
value implied for $N_o$ is indifferent.  However, in certain
classes of models labelled by
$(\epsilon_1,\epsilon_2,\epsilon_3)$, either conditions must be
applied to $\epsilon_k$, or else a tree level dilaton tadpole
correlated with a potential for the geometric moduli are created.
However, in the light of this inconsistency, the tadpoles arising
form the $R$-$R$ sector must be satisfied in order to suppress
anomalies,
\begin{eqnarray}
D_{k;o}^{(NS)}=\epsilon_k32{\rm ,}~D_{k;o}^{(R)}=32 ~{\rm
and}~N_k=D_{k;g}=D_{k;h}=D_{k;f}=0.\nn
\end{eqnarray}

Expanding the amplitude pairs $T+K$ and $A+M$ in powers of $q$, in
correspondence with the gauge group dimensions, one can see that
the coefficients of such terms are integer thereby giving a
consistent particle interpretation.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% New Section
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{Possible Model Classes}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% New Section
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Firstly, we describe the subclass with discrete torsion
$(\epsilon=-1)$. This will necessarily involve negative values for
some or all of the signs of $\epsilon_k$, and as such, $\bar{D}5$
branes are included by construction.  The presence of antibranes
partially breaks supersymmetry so that brane-brane coupling terms
remain supersymmetric, and brane-antibrane configurations have
broken supersymmetry.

In each case the gauge group type can be read off from the charge
structure of the direct annulus and Mobius amplitudes which
manifestly define the form $M^2\pm M$ respectively. So it is
easily seen by these arguments that branes imply a unitary gauge
group, and those of antibranes are unitary simplectic.

In each case, the charges must be parameterized properly so as to
give a structure which is consistent with the unoriented sector.
The annulus must have a charge representation that reflects the
indifference that the Mobius has to orientation. As such, the
parameterization is thus determined by the Mobius and the parts of
the annulus that are eliminated by the tadpole conditions of the
various twisted sectors.

\subsubsection{$\epsilon=(1,1,-1)$ Model}

This configuration corresponds to a $\bar{D}5$ brane wrapped
around the third torus.  With the natural extension of
(\ref{eqn:charsusybreak}) to the $T_{oo}$ sector, the origin terms
in the Mobius read as\footnote{The general Mobius origin is
defined in appendix \ref{app:moborigin}}
\begin{eqnarray}\label{eqn:model_one_mob_origin}
{\cal M}_o=-\frac{1}{8}\big\{-4D_{h;o}\tau^{(-1)}_{oo}+4D_{f;o}\tau_{og}
+4D_{g;o}\tau_{of}+8N_o\tau_{oh}\big\}\nn\\
\end{eqnarray}
which shows that supersymmetry is broken in the $T^{56}$ by the
presence of the $\bar{D}5$ brane in that direction, but $T^{12}$
and $T^{34}$ remain with ${\cal N}=1$ supersymmetry.

It is seen that the only sector in the Mobius that contains $N_o$
branes is $\tau_{oh}$.  The charge representation of the annulus
must be made to agree for each sector.  The direct annulus origin
is determined from (\ref{eqn:dirannorigin}) as
\begin{eqnarray}\label{eqn:annulusorigin}
{\cal A}_o=&\frac{1}{8}&\bigg{\{}\big(2N_o^2+\frac{1}{2}D^2_{k;o}\big)T_{oo}
+2\big{[}N_k^2+D_{k;k}^2+D_{l;k}^2\big{]}T_{ok}\nn\\
&&+2N_oD_{k;o}T^{(\epsilon_k)}_{ko}-4N_kD_{k;k}T^{(\epsilon_k)}_{kk}
+4i(-1)^{k+l}N_lD_{k;l}T^{(\epsilon_k)}_{kl}\nn\\
&&+\frac{1}{2}D_{k;o}D_{l;o}T^{(\epsilon_k\epsilon_l)}_{mo}
-2D_{k;m}D_{l;m}T^{(\epsilon_k\epsilon_l)}_{mm}\nn\\
&&+4i(-1)^{m+k}D_{k;k}D_{l;k}T^{(\epsilon_k\epsilon_l)}_{mk}\bigg{\}}.\nn\\
\end{eqnarray}
Equations (\ref{eqn:model_one_mob_origin}) and
(\ref{eqn:annulusorigin}) highlight the gauge groups as being
$[{U(8)}]_9\times [{U(8)}]_9$ for the $D9$ branes,
$[U(4)]_{5_k}\times [U(4)]_{5_k}$ for the sets of $D5_k$ branes
and $USp(4)^4$ for the $\bar{D}5$.  It will be apparent in further
comments that the matter content and gauge structure are identical
to the $\bb{Z}_2\times \bb{Z}_2$ model in the absence of shifts,
with the effect of a freely acting shift simply halving the
$D_{k;o}$ group and changing the massive spectrum. Such a halving
is necessitated but consistent charge representation. The relevant
charge breaking and factors are displayed in table
\ref{tab:ModelCharges}.
\begin{table}[!ht]
\begin{center}
\begin{tabular}{llllll}
$N_o$ & $=$ & $(n+m+\bar{n}+\bar{m})$, & $N_g$ & $=$ & $i(n+m-\bar{n}-\bar{m})$
\\
$N_f$ & $=$ & $i(n-m-\bar{n}+\bar{m})$, & $N_h$ & $=$ &
$(n-m+\bar{n}-\bar{m})$\\
$D_{g;o}$ & $=$ & $2(o_1+g_1+\bar{o}_1+\bar{g}_1)$, & $D_{f;o}$ & $=$ &
$2(o_2+g_2+\bar{o}_2+\bar{g}_2)$\\
$D_{h;o}$ & $=$ & $2(a+b+c+d)$, & $D_{g;g}$ & $=$ &
$i(o_1+g_1-\bar{o}_1-\bar{g}_1)$\\
$D_{f;f}$ & $=$ & $i(o_2+g_2-\bar{o}_2-\bar{g}_2)$, & $D_{h;h}$ & $=$ &
$a-b-c+d$\\
$D_{g;f}$ & $=$ & $o_1-g_1+\bar{o}_1-\bar{g}_1$, & $D_{g;h}$ & $=$ &
$-i(o_1-g_1-\bar{o}_1+\bar{g}_1)$\\
$D_{f;g}$ & $=$ & $o_2-g_2+\bar{o}_2-\bar{g}_2$, & $D_{f;h}$ & $=$ &
$i(o_2-g_2-\bar{o}_2+\bar{g}_2)$\\
$D_{h;g}$ & $=$ & $a+b-c-d$, & $D_{h;f}$ & $=$ & $a-b+c-d$\\
\end{tabular}
\end{center}
\caption{$\epsilon=(1,1,-1)$ Model
Charges}\label{tab:ModelCharges}
\end{table}
The particular internal arrangement of charges is set to give the
required particle gauge representations, for example, the vector
which is contained in $\tau_{oo}$ should be in the adjoint
bifundamental and relative signs must give rise to positive gauge
couplings in the $ND$ and $DD$ sectors.  With the factor of one
half used, which is implied by the type IIB projection, the open
origin is thus

\begin{eqnarray}\label{egn:AnplusMobzero}
{\cal A}_o+{\cal M}_o&=&(n\bar{n}+m\bar{m}+g_1\bar{g}_1+o_1\bar{o}_1
+o_2\bar{o}_2+g_2\bar{g}_2)\tau_{oo}\nn\\ \nn\\
&&+(n\bar{m}+m\bar{n}+o_1\bar{g}_1+g_1\bar{o}_1+ab+cd)\tau_{og}\nn\\ \nn\\
&&+(nm+\bar{n}\bar{m}+o_2\bar{g}_2+g_2\bar{o}_2+ac+bd)\tau_{of}\nn\\ \nn\\
&&+(\bar{o}_1\bar{g}_1+o_1g_1+o_2g_2+\bar{o}_2\bar{g}_1+ad+bc)\tau_{oh}\nn\\
\nn\\
&&+\frac{\big(a(a+1)+b(b+1)+c(c+1)+d(d+1)\big)}{2}\tau_{oo}^{NS}\nn\\ \nn\\
&&+\frac{\big(a(a-1)+b(b-1)+c(c-1)+d(d-1)\big)}{2}\tau_{oo}^{R}\nn\\ \nn\\
&&+\frac{\big(o_2(o_2-1)+g_2(g_2-1)+\bar{o}_2(\bar{o}_2-1)+\bar{g}_2
(\bar{g}_2-1)\big)}{2}\tau_{og}\nn\\ \nn\\
&&+\frac{\big(o_1(o_1-1)+g_1(g_1-1)+\bar{o}_1(\bar{o}_1-1)+\bar{g}_1
(\bar{g}_1-1)\big)}{2}\tau_{of}\nn\\ \nn\\
&&+\frac{\big(n(n-1)+m(m-1)+\bar{n}(\bar{n}-1)
+\bar{m}(\bar{m}-1)\big)}{2}\tau_{oh}\nn\\ \nn\\
\end{eqnarray}
for the untwisted sector.  Here, the explicit breaking and
simplectic gauge group is seen in $\tau_{oo}$. In the twisted
sector, $95_{1,2}$ and $5_15_2$ remain supersymmetric while
$9\bar{5}_3$ and $5_1\bar{5}_3$ and $5_2\bar{5}_3$ are broken.
This leaves only $\tau_{ok}$ sectors with ${\cal N}=1$
supersymmetry. The supersymmetric chiral content of $95_{1,2}$ and
$5_15_2$ are displayed in Table \ref{fig:95sectors}.
\begin{table}[!ht]
\begin{center}
\begin{tabular}{|l|llll|}\hline
$5_15_2$ & $(4,1;\bar{4},1)$ & $(1,4;\bar{4},1)$ &
$(\bar{4},1;4,1)$ &
$(1,\bar{4};1,4)$ \\
$95_1$ & $(8,1;1,\bar{4})$ & $(1,8;\bar{4},1)$ &
$(\bar{8},1;4,1)$ & $(1,\bar{8};1,4)$ \\
$95_2$ & $(8,1;1,\bar{4})$ & $(1,\bar{8};\bar{4},1)$ &
$(\bar{8},1;4,1)$ & $(1,8;1,4)$ \\ \hline
\end{tabular}
\caption{$95_{1,2}$ and $5_15_2$ Sectors}\label{fig:95sectors}
\end{center}
\end{table}

With the gauge reduction, the Weyl spinors lying in the
$\bar{5}_3\bar{5}_3$ sector, have $(6,1,1,1)$ plus the three
permutations and chiral multiplets in $(4,4,1,1)$ plus the five
other permutations, as can be read off from
(\ref{egn:AnplusMobzero}). Weyl spinors and scalars in the brane
antibrane sector are,
\begin{table}[!ht]
\begin{center}
\begin{tabular}{|l|ll|}\hline
$5_1\bar{5}_3$ & $(\bar{4},1;1,1,4,1)$ & $(1,\bar{4};1,1,1,4)$ \\
& $(1,4;1,4,1,1)$ & $(4,1;4,1,1,1)$ \\
$5_2\bar{5}_3$ & $(4,1;4,1,1,1)$ & $(1,\bar{4};1,1,1,4)$ \\
& $(1,4;1,1,4,1)$ & $(\bar{4},1;1,4,1,1)$ \\
$9\bar{5}_3$ & $(\bar{8},1;4,1,1,1)$ & $(1,\bar{8};1,4,1,1)$ \\
& $(1,8;1,1,4,1)$ & $(8,1;1,1,1,4)$ \\ \hline
\end{tabular}
\caption{Spinors Coupling to $D\bar{5}$}
\end{center}
\end{table}
\begin{table}[!ht]
\begin{center}
\begin{tabular}{|l|ll|}\hline
$5_1\bar{5}_3$ & $(\bar{4},1;1,1,1,4)$ & $(1,\bar{4};1,1,4,1)$ \\
& $(1,4;4,1,1,1)$ & $(4,1;1,4,1,1)$ \\
$5_2\bar{5}_3$ & $(4,1;1,1,1,4)$ & $(1,\bar{4};1,4,1,1)$ \\
& $(1,4;4,1,1,1)$ & $(\bar{4},1;1,1,4,1)$ \\
$9\bar{5}_3$ & $(\bar{8},1;1,4,1,1)$ & $(1,\bar{8};4,1,1,1)$ \\
& $(1,8;1,1,1,4)$ & $(8,1;1,1,4,1)$ \\ \hline
\end{tabular}
\caption{Complex Scalars Coupling to $D\bar{5}$}
\end{center}
\end{table}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% New Section%%%%%%%%%%%%%%%%%%

\subsubsection{Other Models}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

The analysis that has been done so far to show the intact massless
particle spectrum with reference to the $\bb{Z}_2\times \bb{Z}_2$
model with the halving of the $D$5 groups, the procedure is exactly the same
in
all other possible classes.  The other distinct classes are determined by
$\epsilon=(-1,-1,-1)$,
$\epsilon=(1,1,1)$, $\epsilon=(-1,-1,1)$, where the last two are without
discrete torsion.
The brane structure in each is quite different.
Although these models display
interesting
gauge structures, it is unnecessary to restate the
precise group forms as these can be
found in \cite{CAAS}.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% New Section
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{$T^{2}/\bbbb{Z}_2^3$ With Kaluza-Klein and Winding Shifts}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% New Section
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

The extra $\bb{Z}_2$ is now comprised of a product of the
previously studied momenta shifts and a winding shift which acts
as $A_2$, as defined in (\ref{a1a2a3}). This, in a similar fashion
to the action of the momenta shift on the ground state, induces a
factor of $(-1)^n$, and so the torus now consists of sectors
\begin{eqnarray}
T&=&\frac{1}{8}\bigg{\{}|T_{oo}|^2{\Lambda^1}_{m,n}{\Lambda^2}_{m,n}
{\Lambda^3}_{m,n}\big{(}1+(-1)^{m_1+n_1+m_2+n_2+m_3+n_3}\big{)}\nn\\
&&+|T_{ok}|^2{\Lambda^k}_{m,n}\big(1+(-1)^{m_k+n_k}\big)
{\vline\frac{2\eta}{\theta_2}\vline}^4\nn\\
&&+16|T_{ko}|^2{\Lambda^k}_{m,n}{\vline\frac{\eta}{\theta_4}\vline}^4
+16|T_{kk}|^2{\Lambda^k}_{m,n}{\vline\frac{\eta}{\theta_3}\vline}^4 \nn\\
&&+\epsilon(|T_{gh}|^2+|T_{gf}|^2+|T_{fg}|^2+|T_{fh}|^2+|T_{hg}|^2
+|T_{hf}|^2){\vline\frac{8{\eta}^3}{\theta_2\theta_3\theta_4}\vline}^2
\bigg{\}}.
\end{eqnarray}
Here it is seen that the effect of the additional shift in winding
removes states that where present in the previous case of only an
$A_1$ projection.  In the $|T_{oo}|^2$, $|T_{ko}|^2$ and
$|T_{kk}|^2$ sectors, modular invariance requires the application
of three successive $T$ transforms $T^3$ (as can be seen from
(\ref{eqn:Ttransforms})), this induces sign changes for each order
of $T$ which removes all purely massive states from these sectors.
This removal is expected since the Klein would naturally omit pure
massive states in both momenta and winding, and symmetrization
requires their absence. The Klein amplitude, under the usual
reduction of $\Omega$ follows as,
\begin{eqnarray}
{\cal K}=&\frac{1}{8}&\bigg{\{}(P^{1}P^{2}P^{3}\big{(}1
+(-1)^{m_1+m_2+m_3}\big{)}\nn\\
&&+\frac{\big{(}1+(-1)^{m_l+n_k+n_s}\big{)}}{2}P^{l}W^{k}W^{s}T_{oo}\nn\\
&&+2\times16\epsilon_k\bigg[P^k+\epsilon
W^k\bigg]\bigg{(}\frac{\eta}{\theta_4}\bigg{)}^2T_{ko}\bigg{\}}
\end{eqnarray}
and
\begin{eqnarray}
\tilde{\cal
K}=&\frac{2^5}{8}&\bigg{\{}(W^{1}_eW^{2}_eW^{3}_e+W^{1}_oW^{2}_oW^{3}_o\nn\\
&&+\frac{v_l}{2v_kv_m}\big(W^{l}_eP^{k}_eP^{m}_e+W^{l}_oP^{k}_oP^{m}_o\big)
T_{oo}\nn\\
&&+2\epsilon_k\bigg[ W^k_e+\epsilon
P^k_e\bigg]\bigg{(}\frac{2\eta}{\theta_2}\bigg{)}^2T_{ok}\bigg{\}}.
\end{eqnarray}

The zero lattice mode structure is preserved as the case for only
a Kaluza-Klein shift.  However, with the inclusion of the $A_2$
modulation, there is a corresponding lifting of masses in the
direct channel where the $A_2$ actions on the ground state appear
in the transverse channel amplitude.

The Open sector is changed by the various brane couplings that can
now only appear with integer lattice terms.  The additional
winding shift leaves the lattice origin intact, so the untwisted
diagrams have the same multiplicities as for the case with only
Kaluza-Klein shifts (\ref{eqn:AnnulusOrigin}).  The shift
composite now identifies fixed points by
\begin{eqnarray}
(0,0;0,0)\rightarrow(0+\frac{1}{2},0+\frac{1}{2};0+\frac{1}{2},0+\frac{1}{2}).
\end{eqnarray}
So the overall set of sixteen original fixed points under this
identification trivially has the same reduction to eight fixed
points as for a pure Kaluza-Klein shift identification.  The
multiplicity of the twisted origin therefore remains the same. The
transverse amplitude is
\begin{eqnarray}
\tilde{\cal A}=&\frac{2^{-5}}{8}&\bigg{\{}\bigg{(}N_o^2v_1v_2v_3
W^1W^2W^3\frac{\big(1+(-1)^{n_1+n_2+n_3}\big)}{2}\nn\\
&&+\frac{v_k}{2v_lv_s}D^2_{k;o}W^kP^lP^s\frac{\big{(}1+(-1)^{n_k+m_l+m_s}
\big{)}}{2}\bigg{)}T_{oo}\nn\\
&&+2\times2\bigg{[}N_k^2v_kW^k_e+2D_{k;k}^2v_kW^k+2D^2_{l;k}
\frac{P^k}{v_k}\bigg{]}T_{ko}{\bigg{(}\frac{2\eta}{\theta_4}\bigg{)}}^2\nn\\
&&+2N_oD_{k;o}v_kW^k_eT_{ok}^{(\epsilon_k)}{\bigg{(}\frac{2\eta}
{\theta_2}\bigg{)}}^2\nn\\
&&+2\times2N_kD_{k;k}v_kW^k_eT_{kk}^{(\epsilon_k)}{\bigg{(}
\frac{2\eta}{\theta_3}\bigg{)}}^2\nn\\
&&+2\times4N_lD_{k;l}T_{lk}^{(\epsilon_k)}\frac{8{\eta}^3}
{\theta_2\theta_3\theta_4}\nn\\
&&+D_{k;o}D_{l;o}\frac{P^m_e}{v_m}T_{om}^{(\epsilon_k\epsilon_l)}
{\bigg{(}\frac{2\eta}{\theta_2}\bigg{)}}^2\nn\\
&&+2D_{k;m}D_{l;m}\frac{P^m}{v_m}T_{mm}^{(\epsilon_k\epsilon_l)}
{\bigg{(}\frac{2\eta}{\theta_3}\bigg{)}}^2\nn\\
&&+2\times4D_{k;k}D_{l;k}T_{km}^{(\epsilon_k\epsilon_l)}
\frac{8{\eta}^3}{\theta_2\theta_3\theta_4}\bigg{\}}.
\end{eqnarray}
The brane configuration is shown in figure \ref{aZ2xZ2xZ22},
2T-duals have been applied to the coordinates in $T_{4,5}$ and
$T_{67}$.  This rotates $D_{go}$ and $D_{fo}$ to wrap $T_{67}$ and
$T_{4,5}$ respectively. However, $D_{ho}$ is now a $D3$ brane.  The
contribution of the $D9$'s, now $D5^{\prime}$'s is illustrated by
the dashed lines. In a similar fashion to the previous case of the
$A_1$ shift, one has a diagonal pairing of branes.

%%%%%%%%%%%%%%%%%%%%%%Transverse Annulus D5 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}
\centerline{\epsfxsize 2.0 truein \epsfbox {aZ2xZ2xZ22.eps}}
\caption{$D5_{k;o}$ configurations} \label{aZ2xZ2xZ22}
\end{figure}
%%%%%%%%%%%%%%%%%%%%% END OF FIGURE %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

The corresponding direct channel is
\begin{eqnarray}
{\cal A}=&\frac{1}{8}&\bigg{\{}\bigg{(}\frac{N_o^2}{2}\big(P^1P^2P^3
+P^1_{m+\frac{1}{2}}P^2_{m+\frac{1}{2}}P^3_{m+\frac{1}{2}}\big)\nn\\
&&+\frac{1}{2}\frac{D^2_{k;o}}{2}\big(P^kW^lW^m+P^k_{n+\frac{1}{2}}
W^l_{n+\frac{1}{2}}W^m_{n+\frac{1}{2}}\big)\bigg{)}T_{oo}\nn\\
&&+\bigg{[}\frac{1}{2}N_k^2\big(P^k+P^k_{m+\frac{1}{2}}\big)
+2D_{k;k}^2P^k\nn\\
&&+2D^2_{l;k}W^k\bigg{]}T_{ok}{\bigg{(}\frac{2\eta}{\theta_2}\bigg{)}}^2\nn\\
&&+N_oD_{k;o}\big(P^k+P^k_{m+\frac{1}{2}}\big)
T_{ko}^{(\epsilon_k)}{\bigg{(}\frac{\eta}{\theta_4}\bigg{)}}^2\nn\\
&&-2N_kD_{k;k}\big(P^k+P^k_{m+\frac{1}{2}}\big)T_{kk}^{(\epsilon_k)}
{\bigg{(}\frac{\eta}{\theta_3}\bigg{)}}^2\nn\\
&&+2\times2i(-1)^{k+l}N_lD_{k;l}T_{kl}^{(\epsilon_k)}\frac{2{\eta}^3}
{\theta_2\theta_3\theta_4}\nn\\
&&+\frac{1}{2}D_{k;o}D_{l;o}\big(W^m+W^m_{n+\frac{1}{2}}\big)
T_{mo}^{(\epsilon_k\epsilon_l)}{\bigg{(}\frac{\eta}{\theta_4}\bigg{)}}^2\nn\\
&&-2D_{k;m}D_{l;m}W^mT_{mm}^{(\epsilon_k\epsilon_l)}{\bigg{(}
\frac{\eta}{\theta_3}\bigg{)}}^2\nn\\
&&+2\times2i(-1)^{m+k}D_{k;k}D_{l;k}T_{mk}^{(\epsilon_k\epsilon_l)}
\frac{2{\eta}^3}{\theta_2\theta_3\theta_4}\bigg{\}}.
\end{eqnarray}

The various brane and $O$-plane coupling terms are now modified to
those shown in figure (\ref{fig:ModifiedLattices})
\begin{table}[!ht]
\begin{center}
\begin{tabular}{|c|c|}\hline
$\tilde{\cal A}$ and $\tilde{\cal K}$ Plane Diagrams & Lattice Couplings
\\\hline
$D9-D9$ & $W^1W^2W^3\big(1+(-1)^{n_1+n_2+n_3}\big)$ \\
$D5_k-D5_k$ & $W^kP^lP^m\big(1+(-1)^{n_k+m_l+m_m}\big)$ \\
$O9-O9$ & $W^1_eW^2_eW^3_e+W^1_oW^2_oW^3_o$ \\
$O5_k-O5_k$ & $W^k_eP^l_eP^m_e+W^k_oP^l_oP^m_o$ \\ \hline \hline $\tilde{\cal
M}$ plane
diagrams & Lattice Couplings
\\\hline
$D9-O9$ & $W^1_eW^2_eW^3_e+(-1)^{n_1+n_2+n_3}W^1_eW^2_eW^3_e$ \\
$D9-O5_k$ & $W^k_e+(-1)^{n_k}W^k_e$ \\
$D5_k-O9$ & $W^k_e$ \\
$D5_k-O5_k$ & $W^k_eP^l_eP^m_e+(-1)^{n_k+m_l+m_m}W^k_eP^l_eP^m_e$ \\
$D5_k-O5_l$ & $P^m_e$ \\\hline
\end{tabular}
\caption{Modified Lattice Terms}\label{fig:ModifiedLattices}
\end{center}
\end{table}
which gives transverse Mobius as
\begin{eqnarray}
\tilde{\cal M}=-&\frac{1}{4}&\bigg{\{}N_ov_1v_2v_3W^1_eW^2_eW^3_e
\big(1+(-1)^{n_1+n_2+n_3}\big)\hat{T}_{oo}\nn\\
&&+\epsilon_kD_{k;o}\frac{v_k}{2v_lv_s}\big(W^k_eP^l_eP^s_e
+(-1)^{n_k+m_l+m_s}W^k_eP^l_eP^s_e\big)T_{oo}^{(\epsilon_k)}\nn\\
&&+\bigg(\epsilon_kN_ov_k\big(W^k_e+(-1)^{n_k}W^k_e\big)
\hat{\tilde{T}}_{ok}\nn\\
&&+D_{k;o}v_kW^k_e\hat{T}_{ok}^{(\epsilon_k)}\bigg)
{\bigg(\frac{2\hat{\eta}}{\hat{\theta}_2}\bigg)}^2\nn\\
&&+\epsilon_m
D_{l;o}\frac{P^k_e}{v_k}\hat{T}_{ok}^{(\epsilon_l)}
{\bigg(\frac{2\hat{\eta}}{\hat{\theta}_2}\bigg)}^2\bigg{\}}
\end{eqnarray}
which in the direct channel is given by
\begin{eqnarray}
{\cal M}=-&\frac{1}{8}&\bigg{\{}N_o\big(P^1P^2P^3+P^1_{m
+\frac{1}{2}}P^2_{m+\frac{1}{2}}P^3_{m+\frac{1}{2}}\big)\hat{T}_{oo}\nn\\
&&+\epsilon_k\frac{D_{k;o}}{2}\big(P^kW^lW^s+P^k_{m+\frac{1}{2}}W^l_{n
+\frac{1}{2}}W^s_{n+\frac{1}{2}}\big)T_{oo}^{(\epsilon_k)}\nn\\
&&-\bigg(\epsilon_kN_o\big(P^k+P^k_{m+\frac{1}{2}}\big)\hat{T}_{ok}
+D_{k;o}P^k\hat{T}_{ok}^{(\epsilon_k)}\bigg){\bigg(\frac{2\hat{\eta}}{\hat{\theta}_2}\bigg)}^2\nn\\
&&-\epsilon_mD_{l;o}W^k\hat{T}_{ok}^{(\epsilon_l)}
{\bigg(\frac{2\hat{\eta}}{\hat{\theta}_2}\bigg)}^2\bigg{\}}
\end{eqnarray}

The tadpole condition is consistent for a halving of the $N_o$ and
$N_k$ multiplicity factors.  The halving of the $D5$'s is
unchanged from the case with an $A_1$ shift.  The tadpoles thus
have the same form as for the case of the $A_1$ shift.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% New Section
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Model Classes of the $A_2$ Shift}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% New Section
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

The Chan-Paton representations for the $\epsilon=(1,1-1)$ model
are defined in table \ref{tab:ChargesA2}.

\begin{table}
\begin{center}
\begin{tabular}{ll}
$N_o=2(n+m+\bar{n}+\bar{m})$, & $N_g=2i(n+m-\bar{n}-\bar{m})$ \\
$N_f=2i(n-m-\bar{n}+\bar{m})$, & $N_h=2(n-m+\bar{n}-\bar{m})$\\
$D_{g;o}=2(o_1+g_1+\bar{o}_1+\bar{g}_1)$, &
$D_{f;o}=2(o_2+g_2+\bar{o}_2+\bar{g}_2)$\\
$D_{h;o}=2(a+b+c+d)$, & $D_{g;g}=i(o_1+g_1-\bar{o}_1-\bar{g}_1)$\\
$D_{f;f}=i(o_2+g_2-\bar{o}_2-\bar{g}_2)$, & $D_{h;h}=a-b-c+d$\\
$D_{g;f}=o_1-g_1+\bar{o}_1-\bar{g}_1$, &
$D_{g;h}=-i(o_1-g_1-\bar{o}_1+\bar{g}_1)$\\
$D_{f;g}=o_2-g_2+\bar{o}_2-\bar{g}_2$, &
$D_{f;h}=i(o_2-g_2-\bar{o}_2+\bar{g}_2)$\\
$D_{h;g}=a+b-c-d$, & $D_{h;f}=a-b+c-d$\\
\end{tabular}
\caption{$A_2$ Charge Structure}\label{tab:ChargesA2}
\end{center}
\end{table}

The redefinition of the $D9$ charges is required not only for
symmetrization purposes, but the vector can only be made to have
bifundamental representation under these conditions.  This then
produces the same open string configurations as in
(\ref{egn:AnplusMobzero}), with the exception of the halving of
all $D9$ stacks.  The corresponding gauge structure is illustrated
in tables \ref{tab:A2Sector1}, \ref{tab:A2Sector2} and
\ref{tab:A2Sector3}.
%
%
\begin{table}[!ht]
\begin{center}
\begin{tabular}{|l|llll|}\hline
$5_15_2$ & $(4,1;\bar{4},1)$ & $(1,4;\bar{4},1)$ &
$(\bar{4},1;4,1)$ &
$(1,\bar{4};1,4)$ \\
$95_1$ & $(4,1;1\bar{4})$ & $(1,4;\bar{4},1)$ &
$(\bar{4},1;4,1)$ & $(1,\bar{4};1,4)$ \\
$95_2$ & $(4,1;1\bar{4})$ & $(1,\bar{4};\bar{4},1)$ &
$(\bar{4},1;4,1)$ & $(1,4;1,4)$ \\ \hline
\end{tabular}
\caption{$95_{1,2}$ and $5_15_2$ Sectors}\label{tab:A2Sector1}
\end{center}
\end{table}
%
%
Weyl spinors lying in the $\bar{5}_3\bar{5}_3$ sector, have as before,
$(6,1,1,1)$ plus the three
permutations and chiral multiplets in $(4,4,1,1)$ plus the five
other permutations. Weyl spinors and scalars in the brane
antibrane sector are
%
%
\begin{table}[!ht]
\begin{center}
\begin{tabular}{|l|ll|}\hline
$5_1\bar{5}_3$ & $(\bar{4},1;1,1,4,1)$ & $(1,\bar{4};1,1,1,4)$ \\
& $(1,4;1,4,1,1)$ & $(4,1;4,1,1,1)$ \\
$5_2\bar{5}_3$ & $(4,1;4,1,1,1)$ & $(1,\bar{4};1,1,1,4)$ \\
& $(1,4;1,1,4,1)$ & $(\bar{4},1;1,4,1,1)$ \\
$9\bar{5}_3$ & $(\bar{4},1;4,1,1,1)$ & $(1,\bar{4};1,4,1,1)$ \\
& $(1,4;1,1,4,1)$ & $(4,1;1,1,1,4)$ \\ \hline
\end{tabular}
\caption{Spinors Coupling to $D\bar{5}$}\label{tab:A2Sector2}
\end{center}
\end{table}
%
%
\begin{table}[!hb]
\begin{center}
\begin{tabular}{|l|ll|}\hline
$5_1\bar{5}_3$ & $(\bar{4},1;1,1,1,4)$ & $(1,\bar{4};1,1,4,1)$ \\
& $(1,4;4,1,1,1)$, & $(4,1;1,4,1,1)$, \\
$5_2\bar{5}_3$ & $(4,1;1,1,1,4)$ & $(1,\bar{4};1,4,1,1)$ \\
& $(1,4;4,1,1,1)$ & $(\bar{4},1;1,1,4,1)$ \\
$9\bar{5}_3$ & $(\bar{4},1;1,4,1,1)$ & $(1,\bar{4};4,1,1,1)$ \\
& $(1,4;1,1,1,4)$ & $(4,1;1,1,4,1)$ \\ \hline
\end{tabular}
\caption{Complex Scalars Coupling to
$D\bar{5}$}\label{tab:A2Sector3}
\end{center}
\end{table}
%
%
As with the $A_1$ shift, similar extrapolations for the models of
$(1,-1,-1)$, $(-1,-1,-1)$ and $(1,1,1)$ follow from the cases
involved in the $\bb{Z}_2\times \bb{Z}_2$ orbifold model under the
stated breaking of $D9$'s and $D5$'s.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% New Section
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Conclusions}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% New Section
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

In contrast to the models obtained in \cite{CAAS} (and also
similar constructions made in \cite{IAEDAS}), the $A_1$ shift
action on the Kaluza-Klein modes serves not only to give rise to a
change in the internal brane and $O$-plane positions, but also
allows one to break the $D5$ brane structure while keeping that of
the $D9$'s. The choice of modulating by $\bb{Z}_2^3$ with a
Kaluza-Klein shift residing outside the orbifold group, allows the
possibility of partial breaking of supersymmetry to obtain more
detailed models with a richer massless and massive spectrum
through the inclusion of discrete torsion.  This modulation, in
comparison to $\bb{Z}_2\times \bb{Z}_2$ cases without freely
acting shifts, gives rise to a breaking in $SO(8)$ which allows a
much richer counting of states in the Mobius.  This in turn
facilitates an extra breaking of the $D9$ and $D5$ stacks the
Mobius since its topology implies that it should not be
orientable.

As is always the case with the $A_1$ shift, the $D9$'s are
unaffected. Discrete torsion facilitates a far richer counting of
states in all amplitudes. It allows a set of counting in the torus
of the independent orbits that translate to the counting of
$T_{kl}$ states.

In the case of the $A_2$ shift, similar principles apply except the
removal of
purely massive states, effectively reducing the counting of distinct
couplings while
retaining the full complement of world sheet fermionic and bosonic
excitations.

Incorporation the $A_2$ as an additional modulation on top of that of
an orbifold,
facilitates correct particle interpretation for an effective halving of
all $D9$
configurations.  This is unlike the cases studied so far using composite
winding elements
in that previous studies imply the effect of halving takes place on
$N_o$ only.

The distinct shift actions are an interesting way of halving the
group without having to change the background as has been shown in
the case with non vanishing flux of the antisymmetric $NS$-$NS$
field in \cite{C}.  However, the effect of tilting the torus with
such a background proves more interesting as one can break more
precisely according to the rank of the field, and as shown in
\cite{GSH}, obtain extra tensor multiplets in the twisted sector
of the $\bb{Z}_2$ modulated type $IIB$ string.  The effect of both
Kaluza-Klein and winding shifts have been discussed extensively in
the literature (most notably in \cite{CAAS} and \cite{IAEDAS}) in
terms of the spectra and effects on supersymmetry.  A natural step
from here is a study of the type I vacuum with intersecting
branes.  This has been done for the type $IIA$ case without shift
actions in \cite{typeiphun} with supersymmetric spectrum and
$SU(3)\times SU(2)_L\times U(1)_Y$ plus other groups.

The work that is to be done in the type I setting will involve
looking at the open descendants that propagate on non-trivial
backgrounds. It has been shown that magnetic fields can induce
branes to intersect.  This has the result of shifting the modes of
the world sheet bosons and fermions as shown in \cite{AGN}.  This
in combination with an additional antisymmetric background will
lead to a large degree of freedom to control the brane stacks.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% New Section
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Acknowledgments}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% New Section
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

We would like to thank Carlo Angelantonj,
Emilian Dudas and Jihad Mourad for useful
discussions. AF would like to thank the Orsay theory group and
LPTENS for hospitality in the initial phase of this work. This
work is supported in part by the Royal Society and PPARC.

\appendix

\section{Shift Action on Mass}\label{app:massshift}
Here, we show the explicit action of the shift on mass after an
$S$ transformation.

The compact form (\ref{eqn:compact}) is written in contour form
\begin{eqnarray}
(-1)^{m_k}\Lambda_{m_k,n_k}=\frac{1}{2\pi
i}\oint_C\frac{d^2z}{e^{2\pi i z}-1}e^{i\pi \big
(z+\frac{2i\tau_2z^2}{R^2}+2\pi z n \tau_1+i\frac{2\tau_2 n^2
R^2}{4}\big)}.\nn
\end{eqnarray}
For the contours, take
\begin{eqnarray}
(Imz>0),&&C_1~\leftrightarrow~\frac{1}{e^{2\pi i
z}-1}=-\sum^{\infty}_{\tilde{m}}e^{2\pi i\tilde{m}z}\nn\\
(Imz<0),&&C_2~\leftrightarrow~e^{2\pi i z}\frac{1}{e^{2\pi i
z}-1}=\sum^{-1}_{\tilde{m}=-\infty}e^{2\pi i\tilde{m}z}.\nn
\end{eqnarray}
The two contour integrals become
\begin{eqnarray}
\frac{1}{2\pi
i}\int_{-\infty}^{\infty}dz\sum_{\tilde{m}=
-\infty}^{\infty}e^{\big\{-\frac{2\pi
\tau_2}{R^2}\big(z-\frac{iR^2}{2\tau_2}(\tilde{m}+\frac{1}{2}
+n\tau_1)\big)^2+\frac{\pi
R^2}{2\tau_2}(\tilde{m}+\frac{1}{2}+n\tau_1)^2-\frac{\pi\tau_2
n^2R^2}{2}\big\}}.\nn
\end{eqnarray}
By virtue of the gaussian integral, this is thus represented as
\begin{eqnarray}
\frac{R}{\sqrt{2\tau_2}}\sum_{\tilde{m}=-\infty}^{\infty}e^{
-\frac{\pi R^2}{2\tau_2}|\tilde{m}+\frac{1}{2}+n\tau|^2}~\rightarrow~\frac{R}{\sqrt{2\tau_2}}\sum_{\tilde{m}=-\infty}^{\infty}e^{-\frac{\pi
R^2}{2\tau_2}|n+(\tilde{m}+\frac{1}{2})\tau|^2}\nn
\end{eqnarray}
after $S$ transformation which acts on the measure components as
\begin{eqnarray}
S:\left(\matrix{ \tau_1 \cr \tau_2 }\right) ~\rightarrow~
\left(\matrix{-\frac{\tau_1}{|\tau|^2} \cr
\frac{\tau_2}{|\tau|^2}\nn }\right).
\end{eqnarray}
This then shows that the roles $\tilde{m}$ and $n$ are
interchanged as winding and Kaluza-Klein respectively. This then
shows the resulting shift in winding induced by the S transform
involving a Kaluza-Klein phase.

\section{General Mobius Origin for the $A_1$ Shift}\label{app:moborigin}

This is the Mobius origin for all classes of models.  It is
provided as a reference to show how the choice of different
classes results in the change of gauge structure through the signs
that are provided by a given model class.
\begin{eqnarray}
{\cal M}_o=&-\frac{1}{8}&\bigg{\{}\bigg(2N_o(1-\epsilon_1-\epsilon_2
-\epsilon_3)-D_{g;o}(1-\epsilon_1+\epsilon_2+\epsilon_3)\nn\\
&&-D_{f;o}(1+\epsilon_1-\epsilon_2+\epsilon_3)-D_{h;o}(1+\epsilon_1
+\epsilon_2-\epsilon_3)\bigg)\hat{\tau}_{oo}^{NS}\nn\\
&&-\bigg(2N_o(1-\epsilon_1-\epsilon_2-\epsilon_3)-D_{g;o}\epsilon_1(1
-\epsilon_1+\epsilon_2+\epsilon_3)\nn\\
&&-D_{f;o}\epsilon_2(1+\epsilon_1-\epsilon_2+\epsilon_3)-D_{h;o}\epsilon_3(1
+\epsilon_1+\epsilon_2-\epsilon_3)\bigg)\hat{\tau}_{oo}^{R}\bigg{\}}\nn\\
&&+\bigg(2N_o(1-\epsilon_1+\epsilon_2+\epsilon_3)-D_{g;o}(1-\epsilon_1
-\epsilon_2-\epsilon_3)\nn\\
&&-D_{f;o}(-1-\epsilon_1-\epsilon_2+\epsilon_3)-D_{h;o}(-1-\epsilon_1
+\epsilon_2-\epsilon_3)\bigg)\hat{\tau}_{og}^{NS}\nn\\
&&-\bigg(2N_o(1-\epsilon_1+\epsilon_2+\epsilon_3)-D_{g;o}\epsilon_1(1
-\epsilon_1-\epsilon_2-\epsilon_3)\nn\\
&&-D_{f;o}\epsilon_2(-1-\epsilon_1-\epsilon_2+\epsilon_3)-D_{h;o}
\epsilon_3(-1-\epsilon_1+\epsilon_2-\epsilon_3)\bigg)
\hat{\tau}_{og}^{R}\bigg{\}}\nn\\
&&+\bigg(2N_o(1+\epsilon_1-\epsilon_2+\epsilon_3)-D_{g;o}(-1
-\epsilon_1-\epsilon_2+\epsilon_3)\nn\\
&&-D_{f;o}(1-\epsilon_1-\epsilon_2-\epsilon_3)-D_{h;o}
(-1+\epsilon_1-\epsilon_2-\epsilon_3)\bigg)\hat{\tau}_{of}^{NS}\nn\\
&&-\bigg(2N_o(1+\epsilon_1-\epsilon_2+\epsilon_3)-D_{g;o}\epsilon_1
(-1-\epsilon_1-\epsilon_2+\epsilon_3)\nn\\
&&-D_{f;o}\epsilon_2(1-\epsilon_1-\epsilon_2-\epsilon_3)-D_{h;o}
\epsilon_3(-1+\epsilon_1-\epsilon_2-\epsilon_3)\bigg)
\hat{\tau}_{of}^{R}\bigg{\}}\nn\\
&&+\bigg(2N_o(1+\epsilon_1+\epsilon_2-\epsilon_3)-D_{g;o}
(-1-\epsilon_1+\epsilon_2-\epsilon_3)\nn\\
&&-D_{f;o}(-1+\epsilon_1-\epsilon_2-\epsilon_3)-D_{h;o}(1
-\epsilon_1-\epsilon_2-\epsilon_3)\bigg)\hat{\tau}_{oh}^{NS}\nn\\
&&-\bigg(2N_o(1+\epsilon_1+\epsilon_2-\epsilon_3)-D_{g;o}\epsilon_1(-1
-\epsilon_1+\epsilon_2-\epsilon_3)\nn\\
&&-D_{f;o}\epsilon_2(-1+\epsilon_1-\epsilon_2-\epsilon_3)
-D_{h;o}\epsilon_3(1-\epsilon_1-\epsilon_2-\epsilon_3)\bigg)
\hat{\tau}_{oh}^{R}\nn
\end{eqnarray}

%=========================================================================
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\end{document}
%============================================================================
\\
Title:  Open Descendants of NAHE-based free fermionic and Type I Z2^n models
Authors: Dave J. Clements, Alon E. Faraggi
Comments: 42 pages. Standard LaTeX.
Report-no: OUTP--03--04P
\\
The NAHE-set, that underlies the realistic free fermionic models,
corresponds to Z2XZ2 orbifold at an enhanced symmetry point, with
(h_{11},h_{21})=(27,3). Alternatively, a manifold with the same data
is obtained by starting with a Z2XZ2 orbifold at a generic point on the
lattice and adding a freely acting Z2 involution. In this paper we study
type I orientifolds on the manifolds that underly the NAHE-based models by
incorporating such freely acting shifts. We present new models in the
Type I vacuum which are modulated by Z2^n for n=2,3. In the case of n=2,
the Z2XZ2 structure is a composite orbifold Kaluza-Klein shift arrangement.
The partition function provides a simpler spectrum with chiral matter.
For n=3, the two cases discusse are Z2 modulations of the T^6/(Z2 X Z2)
spectrum. The additional projection shows an enhanced closed and open
sector with chiral matter.  The second example involves a modulation
using a product of Kaluza Klein and winding shifts. The winding shift
provides this model with a simpler torus spectrum, particularly with the
removal of part of the twisted sector. In from those which are present in
the Z2XZ2 orbifold. The last two incorporate twisted terms that can provide
discrete torsion, we discuss the spectral content of each model with
discrete torsion with particular choice epsilon=(1,1,-1). 
\\

